*&+ 


. 


SELECTED   TOPICS    IX   THE   THEORY   OF   DIVER 
GENT  SERIES  AND  OF  CONTINUED  FRACTIONS. 

BY   EDWARD  B.  VAN    VLECK. 

PART  I. 

LECTURES  1-4.     DIVERGENT  SERIES. 

IT  may  not  be  inappropriate  for  me  to  preface  the  first  four 
lectures  with  a  few  words  of  a  general  character  concerning  diver 
gent  series.  These  will  serve  the  double  purpose  of  indicating 
the  nature  of  the  problems  to  be  treated  and  of  binding  together 
the  separate  lectures. 

The  problem  presented  by  any  divergent  ser'js  is  essentially  a 
functional  one.  When  a  divergent  series  of  numbers  is  given,  its 
genesis  is  usually  to  be  found  in  some  known  or  unknown  func 
tion.  The  value  which  we  attach  to  it  is  defined  as  the  limit  of 
a  suitably  chosen  convergent  process,  and  the  elements  of  the  proc 
ess  are  the  terms  of  the  given  series  or  are  functions  having  these 
terms  for  their  individual  limits.  Most  commonly  the  given 
numerical  series 

o0  +  «j  +  a2  +  . . . 

is  connected  with  the  power  series 

(1)  a0+  a,x  +  a^2+  •••, 

and  the  question  thus  reduces  to  that  of  determining  under  what 
conditions  or  restrictions  a  value  may  be  assigned  to  the  latter 
series  when  x  approaches  1.  The  primary  topic  therefore  is  the 
divergent  power  series,  and  to  this  we  shall  confine  our  attention 
exclusively. 

This  topic,  if  broadly  considered,  presents  itself  under  at  least 
four  very  different  aspects.  What  is  given  is  in  every  case  a 
power  series  with  a  radius  of  convergence  which  is  not  infinite. 
Suppose  first  that  the  radius  is  greater  than  zero  and  that  the 

781C44 


76 -  -  '     :HEi  BOSTON  COLLOQUIUM. 


*  'circle  of  -convergence  is  -not  a  natural  boundary.  Then  the  series 
defines  within  this  circle  an  analytic  function.  In  the  region  of 
divergence  without  the  circle  the  value  of  the  function  may  be 
obtained  by  the  familiar  process  of  analytic  continuation.  The 
oretically  the  determination  of  the  function  is  a  satisfactory  one, 
for  Poincare  *  has  shown  that  the  function  throughout  the  domain 
in  which  it  is  regular  can  be  obtained  by  means  of  an  enume 
rable  set  of  elements,  Pn(x  —  an).  Practically,  however,  when 
Weierstrass'  process  is  employed  for  analytic  continuation,  the 
labor  is  so  excessive  as  to  render  the  process  nearly  valueless 
except  for  purposes  of  definition.  Hence  to-day  a  search  is  being 
made  for  a  workable  substitute.  I  may  refer  particularly  in  this 
connection  to  the  investigations  by  Borel  and  Mittag-Leffler.  As 
I  consider  the  work  of  the  former  to  be  both  suggestive  and 
practical,  I  have  taken  it  as  the  basis  of  my  second  lecture. 

A  second  aspect  of  our  topic,  intimately  connected  with  the 
continuation  of  the  function  defined  by  (1),  is  the  determination 
of  the  position  and  character  of  its  singularities  in  the  region 
where  the  series  diverges.  This  subject  is  treated  in  Lecture  3. 

When  the  circle  of  convergence  is  a  natural  boundary,  it  does 
not  appear  to  be  impossible,  despite  the  earlier  view  of  Poincare 
to  the  contrary,!  to  discover,  at  least  in  a  certain  class  of  cases, 
an  appropriate,  although  a  non-analytic  mode  of  continuing  the 
function  across  the  boundary  into  other  regions  where  it  will  be 
again  analytic.  The  thesis  of  Borel  and  its  recent  continuation  in  the 
Ada  Mathematica,  together  with  some  excellent  remarks  by  Fabry,\ 
appear  to  be  about  all  that  has  been  done  in  this  direction.  A  very 
brief  discussion  of  the  subject  will  be  given  in  the  fourth  lecture 
in  connection  with  series  of  polynomials  and  of  rational  fractions. 

Lastly,  we  have  the  eonundmm  of  the  truly  divergent  power 
series  —  the  series  which  converges  only  when  x  =  0.  It  is  upon 

*  Rendiconti  del  Circolo  Matematico  di  Palermo,  vol.  2  (1888),  p.  197,  or  see 
Borel' s  Theorie  des  fonctions,  p.  53. 

fThe  conclusions  of  Poincare  and  Borel  are  not  actually  inconsistent,  but  a 
new  point  of  view  is  taken  by  the  latter. 

%Compt.  Rend.,  vol.  128  (1899),  p.  78. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     77 

this  interesting  problem  that  our  attention  will  be  especially 
focused  in  the  first  two  lectures.  In  applying  henceforth  the 
term  divergent  to  power  series,  I  shall  restrict  it  to  series  having 
a  zero-radius  of  convergence. 

I  shall  offer  no  excuse  for  any  irregularity  or  incompleteness  of 
treatment.  The  admirable  treatise  by  Borel  on  Les  Series  diver- 
gentes  (1901)  and  the  masterly  little  book  of  Hadoma rd,  La  Serie 
de  Taylor  et  son  prolongement  analytique  (1901),  leave  little  or  noth 
ing  to  be  desired  in  the  line  of  systematic  development.  While  it 
is  impossible  not  to  repeat  much  that  is  found  in  these  books,  I 
have  also  supplemented  with  other  material  and  sought  to  give  as 
fresh  a  presentation  as  possible. 

LECTURE  1.     Asymptotic  Convergence. 

Few  more  notable  instances  of  the  difference  between  theoretical 
and  practical  mathematics  are  to  be  found  than  in  the  treatment 
of  divergent  series.  After  the  dawn  of  exact  mathematics  with 
Cauchy  the  theoretical  mathematician  shrank  with  horror  from  the 
divergent  series  and  rejected  it  as  a  treacherous  and  dangerous 
tool.  The  astronomer,  on  the  other  hand,  by  the  exigencies  of  his 
science  was  forced  to  employ  it  for  the  purpose  of  computation. 
The  very  notion  of  convergence  is  said  by  Poincare*  to  present  itself 
to  the  astronomer  and  to  the  mathematician  in  complementary  or 
even  contradictory  aspects.  The  astronomer  requires  a  series  which 
converges  rapidly  at  the  outset.  He  cares  not  what  the  ultimate 
character  may  be,  if  only  the  first  few  terms,  twenty  for  example, 
suffice  to  compute  the  desired  function  to  the  degree  of  accuracy 
required.  Consequently  he  judges  the  series  by  these  terms. 
If  they  increase,  the  series  is  for  him  non-convergent.  To  the 
mathematician  the  question  is  not  at  all  concerning  the  nature  of 
the  series  ab  initioj  but  solely  concerning  its  ultimate  character. 

Let  me  illustrate  the  difference  by  referring  to  Vessel's  series 

~»     /  oJ  v* 

j  _  x    / j ?: _j_ 

methodes  nouvelles  de  la  mecanique  cekste,  vol.  2,  p.  1. 


78  THE  BOSTON  COLLOQUIUM. 

which  is  a  solution  of  the  equation 

<*>  *g  +  •£**•--*-  a 

This  is  convergent  for  all  values  of  x,  but  when  x  is  very  large 
the  series  is  worthless  for  computation-lowing  to  the  rapid  and 
long-continued  increase  of  the  terras  before  the  convergence  finally 
sets  in.  The  astronomer  and  physicist  therefore  have  been  driven 
to  use  for  large  values  of  x  an  expansion  which  is  of  the  form  * 

A       A 

A+--  +  -+- 


or,  what  is  the  same  thing, 


cos  : 


(3) 


L  -f  - 


Here  the  multipliers  of  C  and  I)  are  only  formal  solutions  of  the 
differential  equation  (2).  In  respect  to  convergence  they  have  a 
character  exactly  opposite  to  that  of  Jn,  since  for  very  large  values 
of  x  the  terms  at  first  decrease  rapidly  but  finally  an  increase 
begins.  At  this  point  the  computer  stops  and  obtains  a  good  ap 
proximate  value  of  Jn. 

What  is  the  significance  of  this  ?  It  is  strange  indeed  that  no 
attempt  was  made  to  study  the  question  until  1886,  when  Pom-. 
care  f  and  Stidtjes  J  simultaneously  took  it  up.  That  so  evident 
and  important  a  problem  should  have  been  so  long  ignored  by 
the  mathematician  emphasizes  strongly  the  need  of  closer  touch 
between  him  and  the  astronomer  and  the  physicist.  Both  Poineart 
and  Stidtjes  regarded  the  series  as  the  asymptotic  representation 

*See,  for  example,  Gray  and  Mathew's  Treatise  on  Bessel  Functions,  chap.  4. 

f  Acta  Math,,  vol.  8,  p.  295  ff. 

J  Thesis,  Ann.  de  V  EC.  Nor.,  ser.  3,  vol.  3,  p.  201. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.      79 

of  one  or  more  functions.  While  the  latter  writer  studied  care 
fully  certain  divergent  series  of  special  importance  with  the  object 
of  obtaining  from  the  series  a  yet  closer  approximation  to  the 
function  by  a  species  of  interpolation,  Poincare  developed  the 
idea  of  asymptotic  representation  into  a  general  theory. 

To  explain  this  theory  *  and  at  the  same  time  to  develop  certain 
aspects  scarcely  considered  by  Poincart,  I  shall  start  with  the 
genesis  of  a  Taylor's  series.  Take  an  interval  (0,  a)  of  the  posi 
tive  real  axis,  and  denote  by  f(x)  any  real  function  which  is  con 
tinuous  and  has  n  +  1  successive  derivatives  at  every  point  within 
the  interval.  Xo  hypothesis  need  be  made  concerning  the  char 
acter  of  the  function  at  the  extremities  of  the  interval  except  to 
suppose  that  /(a),  /'(#),  •  •  •,  f^(p)ln  \  have  limiting  values  aQJ  «p 
•  .  -,  an  when  x  approaches  the  origin.  Thus  the  function  at  any 
point  within  the  interval  will  be  represented  by  Taylor's  formula  : 


ax 


If  the  function  is  unlimitedly  differentiable  and  limiting  values 
of/(n)(.r)/>i  !  exist  for  all  values  of  n  when  x  approaches  0,  the 
number  of  terms  in  the  formula  can  be  increased  to  any  assigned 
value.  Thus  the  function  gives  rise  formally  to  a  series 

(1)  o()-f  «rT-|-  «2.r2+  ••-, 

uniquely  determined  by  the  limiting  values  of  the  function  and  its 
derivatives. 

The  converse  conclusion,  that  the  series  determines  uniquely  a 
function  fulfilling  the  conditions  above  imposed  in  some  small  in 
terval  ending  in  the  origin,  can  not,  however,  be  drawn.  This  is 
not  even  the  case  when  the  series  is  convergent.  Suppose,  for 
example,  that  an  =  0  for  all  values  of  n.  Then  in  addition  to 

*Cf.  Peano,  Atti  deUa  R.  Accad.  ddle  Scienze  di  Torino,  vol.  27  (1891),  p.  40  ; 
reproduced  as  Anhang  III  ("Ueber  die  Taylor'  sche  Formel")  in  Genocchi- 
Peano's  Differential-  und  Integral-  Rechnung,  p.  359. 


80  THE  BOSTON  COLLOQUIUM. 

i 

f(x)  =  0  we  have  the  functions  e~l/x,  e~l/x\  -  •  •  ,  which  fulfill  the 
assigned  conditions.  They  are,  namely,  unlimitedly  differentiable 
within  a  positive  interval  terminating  in  the  origin,  and  when  x 
approaches  the  origin  from  within  this  interval,  the  functions  and 
their  derivatives  have  the  limit  0.  From  this  it  follows  imme 
diately  that  if  values  other  than  zero  be  prescribed  for  the  an,  the 
function  will  not  be  uniquely  determined,  since  to  any  one  deter 
mination  we  may  add  constant  multiples  of  e~1/x,  e~l/x"~, 

Inasmuch  as  the  correspondence  between  the  function  and  the 
series  is  not  reversibly  unique,  the  series  can  not  be  used,  in 
general,  for  the  computation  of  the  value  of  the  generating  func 
tion.  Nevertheless,  although  this  is  the  case,  the  series  is  not 
without  its  value.  For  consider  the  first  m  terms,  m  being  a 
fixed  integer.  If  x  is  sufficiently  diminished  in  value,  each  of 
these  terms  can  be  made  as  small  as  we  choose  in  comparison  with 
the  one  which  precedes  it,  and  the  series  therefore  at  the  begin 
ning  has  the  appearance  of  being  rapidly  convergent,  even  though 
it  be  really  divergent.  Evidently  also  as  x  is  decreased,  it  has 
this  appearance  for  a  greater  and  greater  number  of  terms,  if  not 
throughout  its  entire  extent.  Now  by  hypothesis  the  generating 
function  was  unlimitedly  differentiable  within  the  interval,  and 
the  successive  derivatives  are  consequently  continuous  within  (0,  a). 
Hence  if  the  interval  is  sufficiently  contracted,  f(m+l\x)  /(m  -f  1)! 
can  be  made  as  nearly  equal  to  am+l  throughout  the  interval  as  is 
desired.  We  have  then  for  the  remainder  in  Taylor's  formula  : 

f(*+WOy\ 

(4)      lUW-V'-au'Ci  +  f) 


in  which  e  is  an  arbitrarily  small  positive  quantity.  Consequently 
if  the  first  m  +  1  terms  of  the  series  should  be  used  to  compute 
the  value  of  the  generating  function,  the  error  committed  would 
be  approximately  equal  to  the  next  term,  provided  x  be  taken  suf 
ficiently  small. 

In  these  considerations  there  is,  of  course,  nothing  to  indicate 
when  x  is  sufficiently  small  for  the  purpose.     If  the  result  holds 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.      81 

simultaneously  for  a  large  number  of  consecutive  values  of  w,  the 
best  possible  value  for  the  function  consistent  with  our  informa 
tion  would  evidently  be  obtained  by  carrying  the  computation 
until  the  term  of  least  absolute  value  is  reached  and  then  stopping. 
Herein  is  probably  the  justification  for  the  practice  of  the  com 
puter  in  so  doing. 

Equation  (4)  which  gave  a  limit  to  the  error  in  stopping  with 
the  (m  -\-  Y)th  term  shows  also  that  this  limit  grows  smaller  as  x 
diminishes.  Since,  furthermore,  by  increasing  m  sufficiently  the 
(wi  +  2)th  term  of  (1)  may  be  made  small  in  comparison  with  the 
(m  -f  l)th  term,  it  is  clear  that  on  the  whole,  as  x  diminishes,  we 
must  take  a  greater  and  greater  number  of  terms  to  secure  the  best 
approximation  to  the  function.  These  two  facts  may  be  comprised 
into  a  single  statement  by  saying  that  the  approximation  given 
by  the  series  is  of  an  asymptotic  character.  This  will  hold 
whether  the  series  is  convergent  or  divergent. 

This  notion  can  be  at  once  embodied  in  an  equation.  From  (4) 
we  have 


(5)  lim 

x=0+ 


=  lim  =  0  (m  =  1,  2,  .  .  .). 

x=0+  « 

This  equation  is  an  exact  equivalent  of  the  two  properties  just 
mentioned  and  is  adopted  by  Poincare  *  as  the  definition  of  asymp 
totic  convergence.  More  explicitly  stated,  the  series  (1)  is  said 
by  him  to  represent  a  function  f(x)  asymptotically  when  equation 
(5)  holds  for  all  values  of  m. 

It  will  be  noticed  that  this  definition  omits  altogether  the 
assumptions  concerning  the  nature  of  the  function  with  which  we 
started  in  deriving  the  series.  Xot  only  has  the  requirement  of 
unlimited  differentiability  within  an  interval  been  omitted  but  the 
existence  of  right-hand  limits  for  the  derivatives  as  x  approaches 
the  origin  is  not  even  postulated.  If  the  value  f/0  be  assigned  to 

*Loc.  cit. 
6 


82  THE  BOSTON  COLLOQUIUM. 

the  function  at  the  origin,  it  will  have  a  first  derivative,  al9  at  this 
point  but  it  need  not  have  derivatives  of  higher  order.* 

The  exclusion  of  the  requirement  of  differentiability  has  un 
doubtedly  its  advantages.  It  enlarges  the  class  of  functions  which 
can  be  represented  asymptotically  by  the  same  series.  It  also 
simplifies  the  application  of  the  theory  of  asymptotic  representa 
tion,  and  this  is  perhaps  the  chief  gain.  The  results  of  Poincare's 
theory  can  readily  be  surmised.  The  sum  and  product  of  two 
functions  represented  asymptotically  by  two  given  series  are 
represented  asymptotically  by  the  sum-  and  product-series  respec 
tively,  and  the  quotient  of  the  two  functions  will  be  represented 
correspondingly,  provided  the  constant  term  of  the  divisor  is  not  0. 
Also  if  f(x)  is  any  function  represented  by  the  series  (1),  whether 
convergent  or  divergent,  and 


is  a  second  series  having  a  radius  of  convergence  greater  than  |  -0  , 
the  asymptotic  representation  of  <£[/(x)]  will  be  the  series  which 
is  obtained  from 

\  +  6iK  +  aix  +  •••)  +  bz(ao  +  aix  +  •  -  -  )2  +  •  •  • 

by  rearranging  the  terms  in  ascending  powers  of  x.  Lastly,  the 
integral  of  f(x)  will  have  for  its  asymptotic  representation  the 
term  by  term  integral  of  (1).  But  the  correspondence  of  the  func 
tion  and  series  may  be  lost  in  differentiation,  for  even  if  the 
function  permits  of  differentiation,  its  derivative  will  not  neces 
sarily  be  a  function  having  an  asymptotic  power  series.  Examples 
of  this  kind  can  be  readily  given.  f 

*  The  ordinary  definition  of  an  nth  derivative  is  here  assumed.     If,  however, 
we  define  the  second  derivative  by  the  expression 


and  the  higher  derivatives  in  similar  fashion,  the  function  must  have  derivatives 
of  all  orders. 

f  Cf.  Borel,  Les  Series  divergences,  p.  35. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.      83 

This  failure  is  on  many  accounts  an  unfortunate  one.  If  a 
further  development  of  Poincare's  theory  is  to  be  made  —  and  this 
seems  to  me  both  a  possibility  and  a  desirability  —  his  definition 
probably  should  be  restricted  by  requiring  (a)  that  the  function 
corresponding  to  the  series  shall  be  unlimitedly  differentiable  in 
some  interval  terminating  in  the  origin,  and  (6)  that  the  deriva 
tives  of  the  function  should  correspond  asymptotically  to  the 
derivatives  of  the  power  series.  These  demands  are  satisfied  in 
the  case  of  an  analytic  function  defined  by  a  convergent  series  and 
seem  to  be  indispensable  for  an  adequate  theory  of  divergent 
series.* 

Thus  far  we  have  considered  asymptotic  representation  only  for 
a  single  mode  of  approach  to  the  origin.  Suppose  now  that  an 
analytic  function  of  a  complex  variable  x  is  represented  by  (1)  for 
all  modes  of  approach  to  the  origin,  and  let  «0  be  the  value  assigned 
to  the  function  at  this  point.  Then  if  the  function  is  one-valued 
and  analytic  about  the  origin,  it  must  also  be  analytic  at  this  point 
since  it  remains  finite.  Hence  the  series  must  be  convergent. 

The  case  which  has  an  interest  therefore  is  that  in  which  the 
asymptotic  representation  is  limited  to  a  sector  terminating  in  the 
origin.  Suppose  then  that  (1)  is  a  given  divergent  series,  and  let 
a  function  be  sought  which  fulfills  the  following  conditions  :  (a) 
the  function  shall  be  analytic  within  the  given  sector  for  values  of 

*  These  requirements  are  formulated  from  a  mathematical  standpoint  with  a 
view  to  extending  the  theory  of  analytic  functions,  and  doubtless  will  be  too 
stringent  for  various  astronomical  investigations.  Prof.  E.  W.  Brown  suggests 
that  for  such  investigations  the  conditions  might  perhaps  be  advantageously 
modified  by  making  the  requirements  for  only  m  derivatives,  771  being  a  number 
which  varies  with  x  and  increases  indefinitely  upon  approach  to  the  critical  point. 
He  also  points  out  the  difficulties  of  an  extension  in  the  case  of  numerous  astro 
nomical  series  which  have  the  form/(x,  t)  =  OQ  -{-  a^x  -j-  o^z2  +  •  •  •,  where  a,-  is  a 
function  of  x  and  t,  cf/dt  being  a  convergent  series.  Poincare's  definition  is  how-  . 
ever  still  applicable. 

Oftentimes  in  celestial  mechanics  the  only  information  concerning  the  func 
tion  sought  is  afforded  in  the  approximation  given  by  the  asymptotic  series.  An 
objection  to  Poincare's  definition  is  that  it  presupposes  a  knowledge  of  the  func 
tion  sought,  for  example,  that  lira  /(*)  =a0,  when  x  =  0.  As  a  matter  of  fact 
the  properties  are  often  unknown.  See  in  this  connection  p.  89  of  these  lectures. 


84  THE  BOSTON  COLLOQUIUM. 

x  which  are  sufficiently  near  to  the  origin  ;  (6)  it  shall  be  repre 
sented  asymptotically  by  the  given  series  within  the  sector,  whether 
inclusive  or  exclusive  of  the  boundary  will  remain  to  be  deter 
mined  ;  (c)  the  asymptotic  representation  shall  not  be  valid  if  the 
angle  of  the  sector  is  enlarged.  So  far  as  I  am  aware,  the  exist 
ence  of  a  function  or  of  functions  which  meet  these  requirements 
has  never  been  demonstrated,  though  it  seems  likely  that  they  in 
general  exist.  It  is,  however,  very  possible  that  the  sector  must 
be  restricted  in  position  as  well  as  in  magnitude.  It  may  be 
found  necessary  to  require  that  the  interior  of  the  sector  shall 
not  include  certain  arguments  of  x  ;  for  example,  in  the  case 
of  the  series  ^m\xm*  the  argument  0,  for  which  the  terms 
have  all  the  same  sign,  f  If  this  be  true,  the  sector  will 
very  probably  have  two  such  arguments  for  its  boundaries. 
When  there  is  a  function  which  satisfies  the  conditions  im 
posed,  it  can  not  be  unique.  For  clearly  e~1/x,  e~1Ae*,  e~l/*5,  •  •  •, 
within  certain  sectors  of  angle  TT,  2?r,  STT,  •  •  •,  have  an  asymptotic 
series  in  which  each  coefficient  is  0.  If,  then,  any  function  has 
been  obtained  satisfying  the  conditions  stated,  one  or  more  of  these 
exponentials,  after  multiplication  by  suitable  constants,  may  be 
added  to  the  function  without  destroying  its  properties.  Hence 
if  a  divergent  series  is  to  represent  a  function  uniquely,  supple 
mentary  conditions  must  be  imposed.  The  nature  of  these  condi 
tions  has  not  yet  been  ascertained.  J 

In  closing  the  general  discussion  a  simple  extension  of  the 
notion  of  asymptotic  convergence  should  be  mentioned  which  is 
necessary  for  the  applications  to  follow.  F(v)  is  said  to  be  repre 
sented  asymptotically  by 


*This  series  is  discussed  in  the  next  lecture. 

t  Borel  (loc.  cit.,  p.  36)  in  his  exposition  of  Poincare's  theory  seems  to  make 
the  definite  statement  that  there  are  arguments  for  which  no  corresponding  func 
tion  exists,  but  I  am  unable  to  find  any  proof  of  the  statement. 

t  In  this  connection  see  pp.  89-92  of  Borel's  article,  Ann.  de  I'  EC.  Nor.,  ser.  3, 
vol.  16  (1899). 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.      85 

when  the  series  in  parenthesis  gives  such  a  representation  of 
F(x)/®(x). 

The  applications  of  Poincare's  theory  have  been  made  chiefly 
in  the  province  of  differential  equations  *  where  divergent  series 
are  of  very  common  occurrence.  We  will  take  for  examination 
the  class  of  equations,  of  which  the  theory  is  perhaps  the  most 
widely  known,  the  homogeneous  linear  differential  equation  with 
polynomial  coefficients  : 

(6)  P»  g  +  Prl(x)  ^  +  •  •  •  +  P0(%  -  0. 

This  is,  in  fact,  the  class  of  equations  to  which  Poincare  first 
applied  his  theory,  f  but  his  discussion  of  the  asymptotic  repre 
sentation  of  the  integrals  was  limited  to  a  single  rectilinear  mode 
of  approach  to  the  singular  point  under  consideration.  The  de 
termination  of  the  sectors  of  validity  for  the  asymptotic  series 
has  been  made  by  Horn,\  who  in  a  number  of  memoirs  has  care 
fully  studied  the  application  of  the  theory  to  ordinary  differential 
equations.  § 

As  is  well  known,  the  only  singular  points  of  (6)  are  the  roots 
of  Pn(x)  and  the  point  x  =  oo.  For  a  regular  singular  point  ||  we 
have  the  familiar  convergent  expressions  for  the  integrals  given 
by  Fuchs.  Consider  now  an  irregular  singular  point.  By  a  linear 
transformation  this  point  maybe  thrown  to  oo,  the  equation  being 
still  kept  in  the  form  (6).  Suppose  then  that  this  has  been  done. 
If  Pn  is  of  the  pth  degree,  the  condition  that  x  =  oo  shall  be  a 
regular  singular  point  is  that  the  degrees  of  Pn_v  -Pn_»  •  -  • ,  PQ 
shall  be  at  most  equal  to  p  —  1,  p  —  2,  •  •  •  ,  p  —  H,  respectively. 

For  an  irregular  singular  point  some  one  or  more  of  the 
degrees  must  be  greater.  Let  h  be  the  smallest  positive  integer 
for  which  the  degrees  will  not  exceed  successively 

*  In  addition  to  the  memoirs  cited  below  Poincare's  Les  methodes  nouvelies  de  la 
mecanique  celeste  and  various  memoirs  by  Kneser  may  be  consulted. 

1[Acta  Math.,  vol.  8  (1886 ),  p.  303.    See  also  Amer.  Jour.,  vol.  7  ( 1885),  p.  203. 

J  Jfoto.  Ann.,  vol.  50  (1898),  p.  525. 

\  See  various  articles  in  Crelles  Journal  and  the  Mathematische  Annalen. 

||  Stelle  der  Bestimmthcit. 


86  THE  BOSTON  COLLOQUIUM. 


The  number  h  is  called  the  rank  of  the  singular  point  oo,  and  the 
differential  equation  can  be  satisfied  formally  by  the  series  of 
Thomae  or  the  so-called  normal  series  : 


(t-1,  2,  ...,n). 

Unless  certain  exceptional  conditions  are  fulfilled,  there  are  n  of 
these  expansions,  and  in  general  they  are  divergent.  To  simplify 
the  presentation  let  us  confine  ourselves  to  the  case  for  which 
h  =  1  .  Then  at  least  one  of  the  polynomials  succeeding  Pn  will 
be  of  the  pth  degree,  and  none  of  higher  degree.  Place 

P  =  Ax*      Ja*-1       •  •  . 


and  construct  the  equation 

(8)  A,*  +  A^sr-\+...  +  At-0. 

The  n  roots  of  this  equation  are  the  n  quantities  a.  which  appear 
in  the  exponential  components  of  the  S.. 

As  a  particular  illustration  of  the  class  of  equations  under  con 
sideration,  BessePs  equation  (  Eq.  (2)  )  may  be  cited.  Here  the 
point  oo  is  of  rank  1,  the  characteristic  equation  is 

AQa*  +  A^  +  A2  =  a2  +  1  =  0, 
with  the  roots 

«i  =  -  *'>     a2  =  +  i> 

and  the  two  Thomaean  integrals  are 


y,  «« 


o  +  V 

/  i> 

-*«*"(  D,  -I-  ^i 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.      87 

in  which  pl9  p2  are  yet  to  be  ascertained.  After  this  has  been 
done,  the  coefficients  of  (9)  can  be  determined  by  direct  substitu 
tion  in  (2). 

To  avoid  complications  we  will  assume  that  the  ?i  roots  of  the 
characteristic  equation  (8)  are  all  distinct,  also  that  the  real  parts  of 
no  two  roots  are  equal.  Mark  now  in  the  complex  plane  the  points 
aiy  a2,  •  •  -,  an,  and  draw  from  them  to  infinity  a  series  of  parallel 
rays  having  such  a  direction  that  no  one  of  the  rays  with  its  pro 
longation  in  the  opposite  direction  shall  contain  two  or  more  of 
these  points.  Finally  surround  the  points  GL  with  small  circles, 


so  that  we  shall  have  the  familiar  loop  circuits  for  the  paths  of 
integration  of  the  integrals  which  we  now  proceed  to  form.     Put 


(10) 


7?i  = 


(i  —  1,  •  ••,  n), 


in  which  vJ(z)  is  a  function  to  be  subsequently  fixed.  In  order 
that  the  integral  may  have  a  sense,  x  will  be  so  restricted  that  the 
real  part  of  zx  shall  be  negative  for  the  rectilinear  parts  of  the 
loop  circuits.  We  can  then  so  determine  v{(z)  that  77.  shall  be  a 
solution  of  (6). 

For  this  purpose  substitute  rj.  for  y  in  (6).     A  reduction,  based 
on  the  integration  of  (10)  by  parts,*  gives  for  r.(z)  the  equation 

dpv 
(11)     (A?  +  A^f-1  +  •  •  •  +  Aa)  -d-p  +..-+()  r  =  0, 

This  is    known  as  Laplace's  transformed  equation.     While  the     • 
original  equation  was  of  the  ?ith  order  with  coefficients  of  the  pth 

*Cf.  Picard's  Traite  cC  Analyse,  vol.  3,  p.  383  ff.,  or  Poincare,  Amer.  Jour., 
vol.  7  (1885),  p.  217  ff. 


88  THE  BOSTON  COLLOQUIUM. 

degree,  the  transform  is  of  the  pth  order  with  coefficients  of  the  nth 
degree.  Its  singular  points  in  the  finite  plane  are  the  roots  of  the 
first  coefficient  of  (11),  which  is  identical  with  the  left  hand  mem 
ber  of  (8).  Furthermore,  an  inspection  of  (11)  shows  immedi 
ately  that  each  of  these  singular  points  a.  is  regular,  and  the 
exponents  which  belong  to  it  are 

0,  1,  2,  -  •  .,  p  -  2,  ^  =  -  (p(  +  1)          (/  =  !,  2,  •  •  .,  «), 

in  which  p.  is  the  exponent  of  x,  hitherto  undetermined  in  (7). 
Hence  if  @.  is  not  an  integer,  there  is  an  integral  of  (11)  having 
the  form 


which,  when  continued  analytically,  can  be  taken  as  the  function 
v{.     Thus  for  the  solution  of  (6)  we  obtain 


If,  finally,  a.  -f  yfx  is  substituted  for  z  the  integral  becomes 
(12)      „,  =  ^ar*-'-*     ey,  (*„  +  k,     +  k       +  •  •  .)dy, 


where  the  transformed  path  of  integration  is  a  loop  circuit  which 
encloses  the  origin  of  the  y-plane,  the  rectilinear  portion  of  the 
path  lying  in  the  half  plane  for  which  the  real  part  of  y  is  negative. 
We  have  thus  reached  a  solution  of  the  differential  equation 
under  the  form  of  an  improper  integral  of  a  convergent  series. 
The  integration  of  (12)  term  by  term,  which  is  a  purely  formal 
process,  gives  at  once  the  normal  integral  St  of  (7),  in  which 


The  asymptotic  character  of  S.  can   be  quickly  demonstrated. 
For  let  unRn(u)  denote  the  remainder  after  n  terms  of  the  series 

£Q  4.  kiU  +  k2u2  +  •  •  •• 
Then 

*Horn,  foe.  cit.,  or  Ada  Math.,  vol.  24  (1901),  pp.  299  ff. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.      89 


Since  the  integral  in  the  right  hand  member,  taken  along  the  loop 
circuit,  can  be  shown  to  remain  finite  when  x  =  ac,  we  have 

lim  *•  Lr~X-»  -     C  +          +..-  +       "    1  =  0. 


But  this  is  the  statement  of  Poincare's  definition  of  asymptotic 
convergence  for  x=  oc. 

I  have  sketched  this  lengthy  process  in  some  detail  because  it 
is  a  thoroughly  typical  one  and  indicates  the  present  status  of  the 
theory  of  asymptotic  series.  It  will  be  observed  that  the  follow 
ing  course  is  pursued  : 

1.  First,  it  is  discovered  that  the  differential  equation  permits 
of  formal  solution  by  a  certain  divergent  series. 

2.  By  some  independent  process  the  existence  of  an  actual  solu-     A 
tion  is  ascertained  which  permits  formally  of  expansion  into  the 
series.     Usually  the  solution  is  found  under  the  form  of  an  inte 
gral,  and  Horn  has  applied  the  theory  chiefly  in  cases  in  which 
solutions  of  this  form  were  known.     (Lately,  however,  he  has 
used    solutions  obtained    from    the  differential   equation  by   the 
process  of  successive  approximation.*) 

3.  The  asymptotic  character  of  the  series  is  then  argued  and, 
finally,  the   sector  within  which  this  representation  is  valid  is 
determined. 

The  status  of  the  theory  thus  exhibited  seems  to  me  an  unsat 
isfactory  and  transitional  one.  It  is  to  be  hoped  that  ultimately 
the  theory  will  be  so  developed  that  the  mere  existence  of  a  diver 
gent  power  series  as  a  formal  solution  of  the  differential  equation 
will  be  sufficient  for  the  immediate  affirmation  of  the  existence  of 
one  or  more  solutions  which  are  analytic  functions  with  certain 
specified  properties. 

*  Math.  Ann.,  vol.  51  (1898),  p.  346.  In  Crelle's  Journal,  vol.  118(1897), 
still  another  method  is  used  for  obtaining  the  solutions. 


90  THE  BOSTON  COLLOQUIUM. 

It  remains  yet  to  fix  the  sectors  within  which  the  solutions  rj. 
can  be  represented  asymptotically  by  the  normal  integrals.  These 
sectors  have  been  specified  by  Horn*  in  the  following  manner. 
Let  straight  lines  be  drawn  from  each  singular  point  a.  to  every 
other  point  and  produce  each  joining  line  to  infinity  in  both  direc 
tions.  A  set  of  lines  will  be  thus  fixed,  radiating  from  the  point  oo . 
Let  their  arguments,  taken  in  the  order  of  decreasing  magnitude, 
be  denoted  by  , 

*>!>    »*•'••!    Wr>   ®r+l   =  *>!  ~  ^    '  '  '  >    W2r  =   ®r  ~  ^ ' 

Suppose  now  that  the  argument  of  the  rectilinear  part  of  the 
path  of  integration  for  ?;a.  in  the  plane  of  z  lies  between  &>p_1  and 
&)p.  Then  TJ.  is  represented  asymptotically  by  8{  for  values  of  the 
argument  of  x  between  Tr/2  —  o)p_1  and  ?r/2  —  ft>p+r.t 

To  the  general  solution  of  (6),  c^  -f  c2?;2  -f  . . .  -f  cnijn,  there 
corresponds  the  divergent  expansion 

(C*  C* 

c> +  ~^r  +  i? 

(13) 


Here  the  real  parts  of  two  exponents,  ax  and  ajxy  are  equal  only 
when  arg(a.  —  0^)0?  is  an  odd  multiple  of  TT/  2;  that  is,  when  argx 
is  equal  to  Tr/2  —  o>.  (i  =  1,  •  •  •  ,  2r).  Suppose  then  that  for 


7T/2  —  a)p_l  <  arg  x  <  Tr/2  — 


p+r 


we  so  assign  subscripts  to  the  a.  that 

jRfoa)  >  R(a2x)  >  .  .  .  >  R(ax). 

Then  all   the  integrals   for  which  cx  =f=  0   have  in  common  the 
asymptotic  series  clSv  while  those  for  which  0^  =  c2=  •  -  •  =  c._v 

*Horn,  Math.  Ann.,  vol.  50  (1898),  p.  531. 

t  In  certain  cases  the  asymptotic  representation  may  be  valid  for  a  greater 
range  of  values  of  the  argument  of  x,  as  in  the  case  of  BesseFs  equation  discussed 
below. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.      91 

c.  =)=  0,  are  represented  by  o^.  Thus  it  appears  that  between  the 
arguments  considered  Sn  is  the  only  one  of  the  n  asymptotic  series 
S.  which  defines  a  solution  of  the  differential  equation  (6)  uniquely. 
Changes  in  the  asymptotic  series  representing  a  solution  may 
occur  from  two  causes,  either  because  x  passes  through  one  of  the 
critical  values  above  mentioned  for  which  there  is  a  change  in  the 
dominant  exponential  in  (13),  or  because  of  a  sudden  alteration  in 
the  values  of  the  constants  ci  for  certain  values  of  the  argument. 
This  can  be  made  clear,  in  conclusion,  by  illustrating  with  Bessel's 
equation.*  For  this  equation,  as  we  saw, 

«i  =  -  <>     «2  =  +  *> 
and  hence 

3?r  TT 


Also  since  Laplace's  transform  for  the  particular  case  before  us  is  f 


the  exponent  p.  for  either  of  the  two  singular  points  z  =  ±  i  has 
the  value  —  J.  Accordingly  the  series  (13)  for  clrjl  -f  c2rj2  may  be 
written 


+  DV(x), 

as  previously  given  in  (3).  If  the  imaginary  part  of  x  is  nega 
tive,  OU(x)  is  the  dominant  term  in  (3)  and  gives  the  asymptotic 
representation  of  the  general  solution,  clrjl  -f  c2?;2.  On  the  other 
hand,  if  the  imaginary  part  is  positive,  the  dominant  term  is 

*  A  brief  but  very  interesting  discussion  is  given  in  a  letter  of  Stokes  in  the 
Ada  Math.,  vol.  26  (1902),  pp.  393-397.  Compare  also  §3  of  Horn's  article, 
Math.  Ann.,  vol.  50  (1898),  p.  525. 

1[Math.  Ann.,  vol.  50,  p.  539,  Eq.  B'  . 


92  THE  BOSTON  COLLOQUIUM. 

D\r(x).  The  changes  in  the  values  of  C  and  D  take  place  only 
when  arg  x  passes  through  the  values  (2n  -f  l)-7r/2.  Then  the 
coefficient  of  the  dominant  term  remains  unaltered,  while  the  coeffic 
ient  of  the  inferior  term  is  altered  by  an  amount  proportional  to  the 
coefficient  of  the  dominant  term.f  We  conclude,  therefore,  that 
in  general  the  asymptotic  series  for  any  solution  of  BesseVs  equa 
tion  changes  abruptly  for  values  of  the  argument  congruent  with 
0  (mod  TT).  Furthermore,  the  series  can  not  be  valid  for  a 
greater  range  of  values  of  the  argument  unless  when  arg  x  =  0, 
either  7)  =  0  or  C  =  0.  In  the  former  case  we  have  a  particular 
solution  Crjl  which  is  represented  by  the  series  CU(x)  for 

—  TT  <  arg  x  <  2-Tr, 

and  in  the  latter  case  a  solution   Drj2  represented  by  DV(x)  for 

—  2-7T  <  arg  x  <  TT. 

I  Thus  from  the  infinitely  many  solutions  of  BesseVs  equation  having 
the  common  asymptotic  representation  CU(x)  and  D  V(x)  respec 
tively,  these  two  solutions  can  be  singled  out  by  the  requirement 

\  that  the  asymptotic  representation  shall  have  the  maximum  sector 
of  validity. 

LECTURE  2.      The  Application  of  Integrals  to  Divergent  Series. 

In  the  first  lecture  a  divergent  series  was  connected  with  a  group 
of  functions,  for  which  it  afforded  a  common  asymptotic  represen 
tation.  In  the  present  lecture  I  shall  treat  of  methods  which 
have  been  used  to  derive  a  function  uniquely  from  the  series. 
To  establish,  whenever  possible,  such  a  unique  connection,  to 
develop  the  properties  of  the  function,  and  to  determine  the  laws 
and  conditions  under  which  the  series  can  be  manipulated  as  a  sub 
stitute  for  the  function  —  this  may  be  said  to  be  the  ultimate  aim 
of  the  theory  of  divergent  series. 

Up  to  the  present  time  this  goal  has  been  reached  only  for  a 
restricted  class  of  divergent  series.  Furthermore,  the  uniqueness 

f  Stokes,  loc.  cit. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.      93 

of  correspondence  between  the  function  and  the  series  has  been 
attained,  not  by  a  specification  of  the  properties  of  the  function, 
but  by  means  of  some  algorithm  which,  when  applied  to  the  series, 
yields  a  single  function.  Unquestionably  the  instrument  by  which 
the  greatest  progress  has  been  made  thus  far  is  the  integral.  The 
first  successes,  however,  were  reached  by  Laguerre  *  and  Stieltjes  f 
through  the  use  of  continued  fractions,  and  very  possibly  in  the 
end  the  continued  fraction  will  prove  to  be  the  best,  as  it  was  the 
earliest  tool.  But  as  yet  it  has  been  applied  only  in  cases  in  which 
the  function  can  be  represented  under  the  form  of  an  integral  as 
well  as  of  a  continued  fraction,  although  with  greater  difficulty. 

To  explain  the   use  of  integrals  let  us   consider  the   familiar 
divergent  series  treated  by  Layuerre, 

(1)  1  +.T  +  2!.^2+  3!a3-f  .... 

This  is,  I  believe,  historically  the  first  divergent  series  from  which 
a  functional  equivalent  was  derived.  J  Since 

*See  No.  20  of  the  bibliography  at  the  end  of  lecture  6. 

t  Bibliography,  No.  26a. 

J  Laguerre  (loc.  cit.)  gives  the  function  first  in  the  form  of  a  continued  fraction    \ 
and  later  proves  its  identity  with  the  integral  which  gives  rise  to  the  divergent    — 
series.     Borel  at  the  opening  of  the  second  chapter  of  Les  Series  divergentes  remarks 
that  "  Laguerre  parait  avoir  le  premier  montre  nettement  Putilite  qu'il  peut  y 
avoir  a  transformer  une  serie  divergente  ...  en  une  fraction  continue  conver- 
gente."     It  seems  almost  to  have  escaped  notice  (see,  however,  p.  110  of  Prings- 
heim's  report,  Encyklopddie  der  Math.  Wissenschaften,  I  A  3),  that  Euler  (Biblio 
graphy,  No.  46  )  derived  a  continued  fraction  from  the  divergent  series 

1  -f  mx  -f  m(m  -f-  n)x2  -f  m(m  +  n)  (m  -j-  2n)x3  -j  ----  , 

of  which  Laguerre'  s  series  is  a  special  case,  and  clearly  realizes  the  utility  of  the 
continued  fraction.  Moreover,  a  close  parallel  to  the  course  followed  by  Laguerre 
is  found  in  the  work  of  Laplace  who  derives  from  the  expression 


a  divergent  series  and  from  this  in  turn  a  continued  fraction,  the  convergents  of 
which  were  stated  by  him  and  proved  by  Jagobi  to  be  alternately  greater  and  less 
than  the  expression.  Had  Jacobi  proved  also  the  convergence  of  the  continued 
fraction,  the  work  of  Laguerre  would  have  had  an  exact  parallel  for  real  values 
of  x.  Cf.  No.  47  of  the  bibliography. 


94  THE  BOSTON  COLLOQUIUM. 

ml  =  T(m  +  1)  =    |     e-*zmdz, 

Jo 

the  series  may  be  written 

rS*X  -»oo 

e~adz  +  x  I     e~~zdz  -f  x2  I     e~zz2dz  -f  •  •  •, 
Jo  Jo 

the  path  of  integration  being  the  positive  real  axis.  If,  then,  by 
a  merely  formal  process,  the  sum  of  the  integrals  is  replaced  by 
the  integral  of  the  sum,  we  obtain 


Je~z  (I  +  xz  +  x2z2  -f-  •  • . )  dz, 
. 

(2)  /(„).»   re-'F(zx),h, 

Jo 


or  a  function 

(2) 

in  which 


1  —  zx 


The  function  thus  derived  is  an  improper  integral  which  has  a 
significance  for  all  values  of  x  except  those  which  are  real  and 
positive.  It  can  be  shown  also  to  be  analytic  for  all  except  the 
excluded  values  of  x.  One  of  the  simplest  proofs  is  as  a  corol- 
•  lary  of  the  following  exceedingly  fundamental  theorem  of  Vallee- 
jPcmssm,*  which  we  shall  have  occasion  to  use  again  later :  If  in 
the  proper  integral 


I 


f(x,z)dz 


the  integrand  is  continuous  in  z  and  x  for  all  values  of  z  upon  the 
path  of  integration  and  for  all  values  of  x  within  a  region  T;  if, 
furthermore,  for  each  of  the  above  values  of  z  it  is  analytic  in  x  over 
the  region  Ty  the  integral  will  also  be  an  analytic  function  of  x  in 
the  interior  of  T.  By  this  theorem,  if  t  is  a  point  on  the  positive 
real  axis, 

e~zdz 


f 

Jo 


—  ZX 


*Ann.  de  la  Soc.  Sclent,  de  JBruxelles,  vol.  17  (1892-3),  p.  323. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.       95 

will  represent  an  analytic  function  of  x  over  any  closed  region  of 
the  ce-plane  which  excludes  the  positive  real  axis.  If,  now,  t 
passes  through  any  indefinitely  increasing  set  of  values,  ' 

'i  <  *2,  <*»•• ', 
we  have  in 


a  series  of  analytic  functions  which  is  seen  at  once  to  converge 
uniformly  over  the  region  considered,  since 

e~sdz 


—  zx 


for  sufficiently  great  values  of  /  andy.  The  limit  (2)  is  therefore 
analytic. 

By  deforming  the  path  of  integration  the  same  conclusion  con 
cerning  the  analytic  character  of 

the  function  (2)  can  be  extended   •  -  f      \  _  _> 

V  /  o  x  * 

to  all  values  of  x  upon  the  posi 

tive  real  axis  excepting  0  and  oo,  and  when  the  deformation  is 
made  on  opposite  sides  of  a  fixed  point  x,  the  two  values  of  the 
integral  will  be  found  to  differ  by 

(3) 

The  integral  accordingly  represents  a  multiple-valued  function 
with  the  singular  points  0  and  oo,  the  various  branches  of  which 
differ  from  one  another  by  multiples  of  the  period  (3).  For  the 
initial  branch  which  was  given  in  (2)  the  limit  of  f^n\x)/nl  will 
be  the  (;i  -f  l)th  coefficient  of  (1)  if  x  approaches  the  origin 
along  any  rectilinear  path  except  the  positive  real  axis. 

Let  the  process  which  has  been  adopted  for  the  series  of  IM- 
guerre  be  applied  next  to  any  other  series 


(I)  «0  +  ajX  -f  a2a 


96  THE  BOSTON  COLLOQUIUM. 

having  a  finite  radius  of  convergence.     If  we  write  the  series  in 
the  form 


then  replace  the  factor  n  !  by  its  expression  as  a  F-integral,  and 
finally,  by  a  step  having  in  general  only  formal  significance,  bring 
all  the  terms  under  a  common  integral  sign,  we  shall  obtain 


f 


j.   \    •  •.».      r 

or 

(4)  f  e-F(zx)dz, 

Jo 

in  which 

This  integral  is  the  expression  upon  which  Borel  builds  his  theory 
of  divergent  series,  and  may  be  regarded  as  a  generalization  of  a 
very  interesting  theorem  of  Caesar  o*  The  series  (5)  is  called 
the  associated  series  of  (I). 

Two  cases  are  now  to  be  distinguished  according  as  the  funda 
mental  series  (I)  has,  or  has  not,  a  radius  of  convergence  R  which 
is  greater  than  0.  If  the  radius  is  not  zero,  the  associated  series 
has  an  infinite  radius  since 


nlan  n  LR-X1  +  €)" 

hm  A  -.  =  lun  \ *—= — -  =  0, 

n=at  \n!      TC=to  \         nl 

and  it  accordingly  represents  an  entire  function.  It  is  a  simple 
matter  to  prove  that  the  integral  (4)  will  have  a  sense  if  x  lies 
within  the  circle  of  convergence  of  (I),  and  that  the  values  of  the 
integral  and  series  are  identical.  But  the  integral  may  also  have 
a  sense  for  values  of  x  which  lie  without  the  circle,  and  in  this  case 
the  integral  may  be  used  to  get  the  analytic  continuation  of  (I). 

*  Cf.  Borel,  Les  Series  divergcntes,  pp.  88-98. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.       97 

The  series  is  said  by  Borel  to  be  swnmable  *  at  a  point  x  when  the 
integral  (4)  has  a  meaning  at  this  point. 

The  second  case  is  that  in  which  the  fundamental  series  is 
divergent.  The  associated  series  in  this  case  may  be  either  con 
vergent  or  divergent.  If  it  is  convergent  only  over  a  portion  of 
the  plane  of  u  =  zx,  we  are  to  understand  by  F(u)  not  merely  the 
value  of  the  associated  series  but  of  its  analytic  continuation. 
Let  x  for  an  instant  be  given  a  fixed  value.  Then  when  z 
describes  the  positive  real  axis,  u  in  its  plane  describes  the  ray 
from  the  origin  passing  through  the  point  x.  If  F(u)  is  holomor- 
phic  along  this  ray,  it  is  possible  that  the  integral  (4)  will  have  a 
sense.  Suppose  that  this  holds  good  as  long  as  x  lies  within  a 
certain  specified  region  of  its  plane.  Then  for  this  region  a  func 
tion  will  be  obtained  uniquely  from  the  divergent  series  by  the  use 
of  the  integral,  precisely  as  in  the  case  of  the  series  of  Laguerre. 

This  method  of  treatment  is  obviously  restricted  to  divergent 
series  for  which  the  associated  series  are  convergent,  and  it  will 
not  always  be  applicable  even  to  these.  A  divergent  series  in  which 
there  is  an  infinite  number  of  coefficients  of  the  same  order  of  mag 
nitude  as  the  corresponding  coefficients  of 

(6)  1  +  x  +  (2  !)V  +  (3  !)2x3  4-  . . .  +  (»!)V  +  . . . 

can  not  be  summed  in  this  manner.  It  will  be  noticed,  however, 
that  the  series  just  given  is  one  whose  first  associated  series  is  the 
series  of  Laguerre,  and  whose  second  associated  series  is  conse 
quently  convergent. 

The  method  of  Borel  can  be  readily  extended  so  as  to  take 
account  also  of  such  series,  or,  more  generally,  of  series  that  have 
an  associated  series  of  the  ?ith  order  which  is  convergent.  One 
mode  of  doing  this  is  by  the  introduction  of  an  n-fold  integral. 
Suppose,  for  example,  that  in  (6)  one  of  the  two  factorials  n !  is 
replaced  by 

e~szndz 


f< 


*  Some  other  term  would  be  preferable  since  his  definition  refers  only  to  one 
of  many  possible  modes  of  summation.  A  series  may  be  simultaneously  "sum- 
mable  "  at  a  point  x  by  one  method,  and  non-summable  by  another. 


98  THE  BOSTON  COLLOQUIUM. 

and  the  other  by 

Xerftit. 
. 

The  (n  -f-  1  )th  term  of  the  series  becomes 

xn  I      I     e-f-zzntndzdt, 

Jo     Jo 

and  we  obtain  the  two-fold  integral 

dzdt 


—  tzx 

for  the  functional  equivalent  of  the  series.  This  is  a  function, 
the  initial  branch  of  which  is  analytic  over  the  entire  plane  of  x 
except  at  the  points  0  and'  oo. 

We  turn  now  to  the  consideration  of  the  region  of  surumability, 
in  which  x  must  lie  in  order  that  the  integral  shall  have  a  sense. 
Borel  has  determined  the  shape  of  this  region  when  the  funda 
mental  series  (I)  is  convergent,  but  in  so  doing  he  restricts  him 
self  to  what  he  calls  the  absolutely  summable  series.  The  series 
(I)  is  said  to  be  absolutely  summable  for  any  value  of  x  when  the 
integral  (4)  is  absolutely  convergent  and  when,  furthermore,  the 
successive  integrals 


have  also  a  sense.* 

To  fix  the  shape  of  the  region  Borel  shows  first  that  if  a  func 
tion  defined  by  a  convergent  series  (I)  is  absolutely  summable  at 
a  point  -P,  it  is  analytic  within  the  circle  described  upon  the  line 
OP  as  diameter,  connecting  P  with  the  origin  0  ;  conversely,  if  it 
is  analytic  within  and  upon  a  circle  having  OP  as  diameter,  it 
must  be  absolutely  summable  along  OP,  inclusive  of  the  point 

*The  condition  (7)  was  not  originally  included  in  Borel's  definition  of  abso 
lute  summability  (Ann.  de  V  EC.  Nor.,  ser.  3,  vol.  16,  1899),  and  is  superfluous 
in  fixing  the  shape  of  the  region.  Cf.  Math.  Ann.,  vol.  55  (1902),  p.  74.  The 
modification  of  the  definition  was  introduced  in  the  Series  diveryentes  and  is 
needed  for  the  developments  explained  below,  p.  102.  Chapters  3  and  4  of  this 
treatise  can  be  read  in  connection  with  the  present  lecture. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.      99 

P.  As  P  moves  outward  from  the  origin  along  any  ray,  the  lim 
iting  position  for  the  circle  is  one  in  which  it  first  passes  through 
a  singular  point  S,  and  at  this  point  SP  and  OS  subtend  a  right 
angle.  The  region  of  absolute  summability  can  therefore  be 
obtained  as  follows  :  Mark  on  each  ray  from  the  origin  the 
nearest  singular  point  of  the  function  defined  by  (I),  if  there 
is  such  a  point  in  the  finite  plane.  Then  through  this  point 
draw  a  perpendicular  to  the  line.  Some  or  all  of  these  per 
pendiculars  will  bound  a  polygon,  the  interior  of  which  con 
tains  the  origin  and  is  not  penetrated  by  any  one  of  the  perpen 
diculars.  This  region  is  called  the  polygon  of  summabiUty.  If 
the  singularities  of  the  function  are  a  set  of  isolated  points,  the 
polygon  will  be  rectilinear.  For  the  extreme  case  in  which  the 
circle  of  convergence  is  a  natural  boundary,  the  polygon  and 
circle  coincide.  In  every  other  case  the  circle  is  included  in  the 
polygon.  Thus  by  the  use  of  (4)  Borel  effects  an  analytic  con 
tinuation  of  the  series  over  a  perfectly  definite  region  whenever  an 
analytic  continuation  exists.  On  passing  to  the  exterior  of  the 
polygon  the  series  ceases  to  be  absolutely  summable.  As  an 
example  of  this  result,  take  the  series 

x3       x5 


which  is  the  familiar  expansion  of  J  log  (1  +  x)/(l  —  x).  The 
singular  points  of  the  function  are  +  1  and  —  1,  the  circle  of 
convergence  is  the  unit  circle,  and  the  polygon  of  summability 
is  a  strip  of  the  plane  included  between  two  perpendiculars  to  the 
real  axis  through  the  points  ±  1. 

When  the  given  series  is  divergent,  the  form  of  the  domain  of 
summability  has  not  been  determined  with  such  precision.     The  / 
only  information  which  we  have  upon  the  subject  is  contained  in  J 
a  brief  but  important  communication  by  Phragmen  in  the  Comp- 
tes    Rendus,*    published    since    the  appearance  of  BoreVs  work. 
Phragmen  considers  here  the  domain,  not  of  absolute,  but  of  sim 
ple  summability  for  Laplace's  integral 

*  Vol.  132,  p.  1396  ;  June,  1901. 


100  THE  BOSTON  COLLOQUIUM. 

(8) 


in  which  f(zx)  denotes  an  arbitrary  function. 

To  adopt  a  term  of  Mittag-Leffler,  the  domain  is  a  "  star," 
which  is  derived  as  follows  :  Draw  any  ray  from  the  origin.  If 
the  series  is  summable  at  a  point  XQ  of  this  line,  Phragmen  shows 
that  it  is  summable  at  every  point  between  x0  and  the  origin  0. 
There  is  therefore  some  point  P  of  the  line  which  separates  the 
interval  of  summability  from  the  interval  of  non-summability. 
If  the  function  is  summable  for  the  entire  extent  of  the  ray,  P 
lies  at  infinity.  In  any  case  let  the  segment  OP  be  obliterated 
and  then  make  a  cut  along  the  remainder  of  the  line.  When  the 
same  thing  is  done  for  every  ray  which  terminates  at  the  origin, 
there  is  left  a  region  called  a  star,  bounded  by  a  set  of  lines  radi 
ating  from  a  common  center,  the  point  at  infinity. 

Phragmen  says  that  the  proof  of  this  result  is  so  simple  that  it 
can  be  given  "  en  deux  mots"  For  this  reason  I  shall  repro 
duce  it  here.  We  are  to  show  that  if  the  integral  converges  for 
any  value  x  =  XQJ  it  will  also  converge  for  x  —  6xQ,  if  0  <  6  <  1  . 
Place 

/«)  =  </>(*)  +  i+(z). 

For  x  =  x0  the  real  and  imaginary  components  of  the  integrals, 
(9)  r#*)e-'efe,       i  r^(z)e-*dz, 

•Jo  Jo 

have  a  sense.     We  are  to  prove  that  the  integrals 


obtained  by  replacing  XQ  by  &XQ,  also  exist.     Consider  either  inte 
gral,  for  example  the  former.     Let  0  <  ax  <a2  <  oc,  and  put 


J= 


DIVERGENT  SERIES  AND  CONTINUED -F^ ACTIONS.   101  , 
By  the  change  of  variable  w  =  Qz  this  becomes 

<j)(w)e~wdw. 


Since  e~^/e~l}  is  a  positive  and  decreasing  function  in  the  interval     , 
considered,  the  second  mean-value  theorem  of  the  integral  calculus* 
may  be  applied,  giving 


(11) 


. 


«)    /»0a 
J0a 


in  which  a  designates  an  appropriate  value  between  at  and  a2. 
This,  as  Phragmen  says,  proves  the  theorem,  but  a  word  or  two 
of  explanation  additional  to  his  "  deux  mots  "  may  not  be  unac 
ceptable  to  some  of  my  hearers.  The  necessary  and  sufficient 
condition  for  the  existence  of  the  first  of  the  two  integrals  given 
in  (10)  is  that  by  taking  two  values  ax  and  a2  sufficiently  small  or 
two  values  sufficiently  large,  the  integral  /  may  be  made  as  small 
as  we  choose.  Xow  this  is  true  of 


r 

t/«o 


<f)(v:)e-irdw 

since  the  integrals  (9)  exist,  and  equation  (11)  show  then  that  it 
must  be  true  likewise  of  J  because  the  factor  e~a^l~0) /O  has  an 
upper  limit  for  0  <  Ql  <  0  <  1  and  0  <  ax  <  oo .  It  follows 
therefore  that  the  integrals  (10)  exist. 

Two  other  facts  stated  by  Phragmen  are  also  of  interest.  The 
function  of  x  defined  by  (8)  is  a  monogenic  function  which  is  holo- 
morphic  at  every  point  in  the  interior  of  a  circle  described  upon 
OP  as  diameter.  If,  also,  in  place  of  f(zx)  we  take  the  associated 
series  F(zx)  of  a  convergent  series  (I),  the  star  of  convergence 
coincides  with  BorePs  polygon  of  absolute  summability.  Thus 
the  regions  of  absolute  and  non-absolute  summability  are  the 
same,  or  differ  at  most  only  in  respect  to  the  nature  of  the  boun 
dary  points. 

*  Bonnet's  form  :  Encyklopadie  der  Math.  Wiss.,  II  A  2,  \  35. 


THE  BOSTON  COLLOQUIUM. 


It  might  be  thought  that  the  result  of  Phragmen  makes  the  con 
cept  of  absolute  summability  useless.  This  is,  however,  in  no 
wise  the  case.  At  any  rate,  Borel  employs  the  concept  to  estab 
lish  the  important  conclusion  that  a  divergent  series,  if  absolutely 
summable,  can  be  manipulated  precisely  as  a  convergent  series. 
Thus  if  two  absolutely  summable  series,  whether  convergent  or 
divergent,  are  multiplied  together,  the  resultant  series  will  also  be 
absolutely  summable,  and  the  function  which  it  defines  will  be 
the  sum  or  product  of  the  functions  defined  by  the  two  former 
series.  Or,  again,  if  an  absolutely  summable  series  is  differen 
tiated  term  by  term,  another  such  series  is  obtained,  and  the  latter 
yields  a  function  which  is  the  derivative  of  the  one  defined  by 
the  former  series.  Lastly,  the  function  determined  by  an  abso 
lutely  summable  series  can  not  be  identically  zero,  unless  all  the 
coefficients  of  the  series  vanish. 

These  facts  make  possible  the  immediate  application  of  Borel'  s 
theory  to  differential  equations.     If,  in  short, 
P(x,  y,  /,-..,  y<»>)  =  0 

is  a  differential  equation  which  is  holomorphic  in  x  at  the  origin  and 
is  algebraic  in  y  and  its  derivatives,  any  absolutely  summable 
series  (I),  which  satisfies  formally  the  equation,  defines  an  analytic 
function  that  is  a  solution  of  the  equation.  For  example,  it  will  be 
found  that  the  series  of  Laguerre  satisfies  formally  the  equation 


and  hence  the  function 

>  -    7 

e~"dz 
1  —  zx 

must  be  a  solution  of  the  equation. 

These  conclusions  of  Borel  should  be  strongly  emphasized. 
In  any  complete  theory  of  divergent  series  it  is  an  ultimatum 
that  they  shall  in  all  essential  points  *  permit  of  manipulation 

*In  an  absolutely  summable  series  it  is  not  always  legitimate  to  change  the 
order  of  an  infinite  number  of  terms.  Cf.  Borel,  Journ.  deMath.,  ser.  5,  vol.  2 
(1896),  p.  111. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    103 

precisely  as  convergent  series,  this  property  being  a  requisite  for 
satisfactory  application  to  differential  equations. 

In  our  preceding  exposition  of  BoreVs  theory,  we  have  intro 
duced  his  chief  integral  by  a  method  which  permits  of  expansion 
in  various  directions.  Le  Roy  in  his  very  excellent  thesis  *  suggests 
a  change  of  the  function  in  Laplace's  integral  which  greatly  en 
larges  the  applicability  f  of  BorePs  method  without  essentially 
changing  its  character.  Let  the  initial  series  (I)  be  first  written 


and  then  replace  the  second  factor  in  each  term  by 
T(np  + 


P     o 

This  gives  for  the  formal  equivalent  of  the  series  the  integral 
(12)  -  f  e-/pzl/p-lF(z.r)dz, 

P  Jo 

in  which  the  associated  function  is  now 


The  number  p  remains  to  be  fixed.     If  the  series  (I)  is  divergent, 
there  is  a  critical  value  of  p  such   that  any  smaller  value  of  p   \- 
gives  an  associated  series  having  a  zero  radius  of  convergence,  } 
while  a  larger  value  gives  one  with  an   infinite  radius  of  conver-  / 
gence.     This  critical  value  p'  may  be  said  to  gauge  or  measure 

*  Annalesde  Toulouse,  ser.  2,  vol.  2  (1900),  p.  416. 

t  Since  this  was  written,  a  very  interesting  application  of  Le  Roy's  idea  to 
differential  equations  has  been  made  by  Maillet,  Ann.  de  V  EC.  Nor.,  ser.  3,  vol. 
20  (1893),  p.  487  ff. 


104  THE  BOSTON  COLLOQUIUM. 

the  degree  of  divergence  of  the  series.  For  the  divergent  series 
treated  by  Borel,  p'  =  1.  If  p'  =  0,  the  series  (I)  has  a  finite 
radius  of  convergence.  On  the  other  hand,  when  p'  =  oo,  Le  Roy's 
integral  can  not  be  applied,  but  it  may  be  conjectured  that  such 
cases  will  be  of  very  rare  occurrence.  Le  Roy  proposes  to  employ 
the  integral  when  the  associated  series  is  convergent  for  p  =  p'  and 
when  also  its  circle  of  convergence  has  a  finite  radius  and  is 
not  a  natural  boundary.  The  function  obtained  from  (12)  will 
be  unique,  and  he  shows  that  the  series  which  are  summable  by 
its  use  like  the  series  of  Bord,  can  be  manipulated  as  convergent 
series.  One  might  also  inquire  whether,  in  case  (13)  diverges  for 
p  =  p'  and  we  take  p  >  p',  we  shall  not  get  a  unique  result  irre 
spective  of  the  value  of  p. 

Other  forms  of  integrals  may  also  be  selected  for  the  summa 
tion  of  the  series,  as  for  example,* 


f(z)F(zx)dz, 


F(zx) 


To  generate  the  given  series  (I)  we  must  so  select/  (x)  and  F(zx) 
that 


Borel  chooses  for  f(z)  the  exponential  function,  making  in  conse 
quence  F(zx),  his  associated  series,  dependent  only  upon  the  given 
series.  Hence  his  process  is  called  very  appropriately  the  ex 
ponential  method  of  summation.  Stieltjes,  f  on  the  other  hand, 
with  his  continued  fraction  arrives  at  an  integral  in  which  F(u)  is 
the  fixed  function  and/  (3)  is  the  variable  function  dependent  on 
the  series  given.  For  the  fixed  function  he  takes 

F(zx)  =  —      -  =  1  +  *x  +  z2x2  +  .  •  ., 


*Cf.  Le  Boy,  toe.  cit.,  pp.  414-415. 
t  Loc.  cit. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    105 

so  that 

(14)  a.-   f 

Jo 


•zndz. 


At  first  sight  this  choice  of  functions  would  seem  to  be  a  very  desir 
able  one,  for  the  function  defined  by  the  divergent  series  is  obtained 
in  the  familiar  form 


Upon  examination,  however,  it  turns  out  to  be  otherwise.  For 
suppose  the  divergent  series  to  be  given  and  f(z)  is  to  be  found. 
The  problem  is  then  a  very  difficult  one,  that  of  the  inversion  of 
the  integral  (14)  when  an  is  given  for  all  values  of  n.  This  is 
what  Stieltjes  terms  "  the  problem  of  the  moments."  It  does  not 
admit  of  a  unique  solution,  for  Stieltjes  himself*  gives  a  function, 

f(z)  —  e~v  2  sin  \/z, 

which  will  make  an=  0  for  all  values  of  n.  If  the  supplementary 
condition  is  imposed  that  f(z)  shall  not  be  negative  between  the 
limits  of  integration,  only  a  single  solution  f(z)  is  possible,  but 
the  divergent  series  is  thereby  restricted  to  belong  to  that  class 
which  Stieltjes  derives  naturally  and  elegantly  by  the  considera 
tion  of  his  continued  fraction. 

Thus  far  our  attention  has  been  confined  exclusively  to  integrals 
in  which  one  of  the  limits  of  integration  is  infinite.  There  are, 
however,  advantages  in  using  appropriate  integrals  having  both 
limits  finite,  at  least  if  the  given  series  is  convergent  and  the 
integral  is  used  for  the  purpose  of  analytic  continuation.  In 
particular,  the  integral 

(16)  .          /(,-) 


should   be  noted,  to  which  Hadamard  has  drawn  attention  in  his 
thesis.  f     This  falls  under  Vallee-Poussin's  theorem  when  V(z)  is 

*&*.«*.,§  55. 

f  Journ.  de  Math.,  ser.  4,  vol.  8  (1892),  pp.  158-160. 


106  THE  BOSTON  COLLOQUIUM. 

continuous  along  the  path  of  integration  and  when  also  F(u)  is 
analytic  in  u  =  zx  for  all  values  of  z  upon  the  path  of  integration 
and  for  values  of  x  in  some  specified  region  of  the  x-plane.  If, 
as  we  suppose,  the  path  is  rectilinear,  the  values  of  x  to  be  ex 
cluded  are  evidently  those  which  lie  on  the  prolongations  of  the 
vectors  from  the  origin  to  the  singular  points  of  F(x).  The 
region  of  convergence  of  (16)  is  consequently  a  star,  whose  boun 
dary  consists  of  prolongation  of  these  vectors.*  Thus  Hadamard's 
integral,  when  applied  to  the  analytic  continuation  of  a  function, 
is  superior  to  Borel's  in  the  extent  of  its  "region  of  summability." 
This  is  illustrated  in  Le  Roy's  thesis  f  with  the  very  familiar  series  : 


r  ~      2-4 2/i 

Here  the  coefficient  of  xn  is 

zndz 


Y/z(\  -  Z) 
so  that 

/(.<•)=: r'     "z 


Since  F(zx)  =  1/(1  —  zx),  the  region  of  summability  is  the  entire 
plane  of  x  with  the  exception  of  the  part  of  the  real  axis  between 
x  =  1  and  x  =  oo.  BorePs  polygon  of  summability  for  the  series, 
on  the  other  hand,  is  only  the  half  plane  lying  to  the  left  of  a 
perpendicular  to  the  real  axis  through  the  point  x  =  1. 

Much,  it  seems  to  me,  can  yet  be  done  in  following  up  the  use 
of  Hadamard's  integral.  One  special  case  has  been  studied  already 
by  Le  Roy,  in  which  the  (n  4-  l)th  coefficient  of  (I)  has  the  form 


«.-  f 

Jo 


pi 

*  This  conclusion  also  holds  if  only  /    V(z)dz  is  an  absolutely  convergent  inte 
gral,  as  is  shown  by  Hadamard. 
f  pi  411. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    107 
The  series  therefore  defines  a  function 


XI ^ 
1          ,JX 


which  is  analytic  over  the  entire  plane  except  along  the  real  axis 
between  x  =  1  and  x  =  oo.  The  path  of  integration  may  also 
permit  of  deformation  so  as  to  show  that  the  cut  between  the 
points  is  not  an  essential  cut.  It  is  interesting  to  note  that  if 
<j)(z)  is  positive  between  0  and  1,  the  primary  branch  of  the  func 
tion  has  only  real  roots  which  are,  moreover,  greater  than  1.* 

LECTURE  3.  On  the  Determination  of  the  Singularities  of  Func 
tions  Defined  by  Power  Series. 

Up  to  the  present  time  comparatively  little  successful  work  has 
been  done  in  determining  the  singularities  of  functions  defined  by 
power  series,  and  the  little  which  has  been  done  relates  mostly  to 
singularities  upon  the  circle  of  convergence.  Work  of  this  special 
nature  I  shall  omit  from  consideration  here,  thus  passing  over  the 
memoirs  of  Fabry,  and  I  shall  call  your  attention  to  the  literature 
which  treats  of  the  singularities  in  a  wider  domain. 

The  most  fundamental  and  practical  result  yet  obtained  is 
undoubtedly  a  brilliant  theorem  of  Hadamardfi  in  the  wake  of 
which  a  number  of  other  interesting  memoirs  have  followed. 
This  theorem  is  as  follows  : 

If  two  analytic  functions  are  defined  by  the  convergent  power  series 

(1)  <£(»)  =  «0  -f  «!«  +  a2x2  +  •  •  -, 

(2)  +(*)-*,  +  bp  +  bf*+—, 

the  only  singularities  of  the  function  ^ 

(3)  fix)  =  a060  +  afrx  +  a&x2  +  .  •  - 

loill  be  points  whose  affixes  7^.  are  the  product  of  affixes  of  the  singu 
lar  points  ai  and  0.  of  the  first  two  functions. 

*Le  Roy,  toe.  cit.,  pp.  330-331. 
"Me/a  Math.,  vol.  22  (1898),  p.  55. 


108  THE  BOSTON  COLLOQUIUM. 

The  possibility  that  x  =  0  should,  in  addition,  be  a  singular 
point  has  been  pointed  out  since  by  Lindelof. 

Although  Hadamard's  proof  of  the  theorem  is  not  a  compli 
cated  one,  I  shall  present  here  a  still  simpler  proof  given  by  Borel* 
Let  R  and  R'  be  the  radii  of  convergence  of  (1)  and  (2)  respec 
tively,  and  take  a  number  p  such  that  R/p>l/R'.  If  then 
|  *x  |  ^  |  px  I  <  R  and  x  >  I/R',  the  product  of  $(zx)  and 
ty(l/x)  can  be  developed  into  a  Laurent's  power  series  which  is 
valid  in  a  circular  ring  in  the  x-plane,  having  its  center  at  the 
origin  and  the  outer  and  inner  radii  R/p  and  l/R'  respectively. 
In  this  product  the  absolute  term  is  obviously 

(4)  A*)  =  a0^  + "A* +  «&*'+•••. 

Consider  now  the  integral 


ill  which  c  is  a  closed  path  surrounding  the  origin  and  contained 
within  the  circular  ring.  As  long  as  z  in  its  plane  lies  within  a 
circle  of  radius  p  <  RR' ',  having  its  center  in  the  origin,  the 
integral  will  surely  define  a  function  of  z,  and  this  function  is 
evidently  equal  to  the  residue  of  the  integrand  for  x  —  0,  which 
is  /(*). 

We  shall  now  seek  to  extend  this  function  by  varying  z  and  at 
^  the  same  time  deforming  appropriately  the  path  of  integration.  By 
^bhe  theorem  of  Vallee  Poussin  quoted  in  Lecture  2,  the  integral 
will  continue  to  represent  an  analytic  function  of  z,  provided  at 
every  stage  the  integrand  remains  analytic  in  x  and  z ;  x  being 
any  point  upon  the  path  of  integration.  Now  the  values  to  be 
avoided  are  clearly  the  singular  points  of  the  functions  <l>(zx)  and 
I ;  namely  the  points  : 


*JBull.  de  la  Soc.  Math,  de  France,  vol.  26  (1898),  pp.  238-248. 

An  interesting  proof  ' '  in  multi  case  "  is  given  without  the  use  of  integrals  by 
Pincherle  in  the  Rendiconto  delta  R.  Acead.  delle  Scienze  di  Bologna,  new  ser.,  vol. 
3  (1*98-9),  PP-  67-74. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    109 


ZX  =£.,       X  = 


1 

K 


The  points  x  =  1  'ft.  lie  within  the  circle  (1/-R')  which  is  the  inner 
circumference  of  the  ring,  while  the  points  x  =  OL.'Z  before  the 
variation  of  z  lie  without  the  outer  circumference  (R/'p).  For 
simplicity  of  presentation  it  may  be  convenient  to  assume  at  first 
that  these  points  form  an  aggregate 
of  isolated  points.  Suppose  then 
that  z  follows  any  path  in  its  plane 
emerging  from  the  circle  (/>).  Then 
the  points  aJz  describe  certain  cor 
responding  paths  which  we  will 
mark  in  the  cc-plane.  At  the  same 
time  the  contour  c  may  be  deformed  ff*/z 
continuously  so  as  to  recede  before 
the  points  a.!z  without  sweeping 
over  any  point  1//3.,  provided  merely  that  ajz  never  collides  with 
a  point  1//9. ;  that  is,  z  must  never  pass  through  a  point  afi.. 
Now  when  z  is  held  fixed,  a  deformation  in  the  contour  c,  subject 
of  course  to  the  condition  indicated,  produces  no  change  in  the  value 
of  the  integral  f(z),  since  the  integrand  is  holomorphic  between 
the  initial  and  deformed  paths.  On  the  other  hand,  when  the 
path  is  kept  fixed  and  z  is  varied,  we  have  the  analytic  continu 
ation  of  tf\z)  in  accordance  with  the  theorem  of  Vallee  Poussin. 
By  the  two  changes  togethei\/(s)  may  be  continued  over  the  entire 
plane  of  z  with  the  exception  of  the  points  a./Qy  =  7^..  To  these 
should,  of  course,  be  added  z  —  oc,  also  z  =  0  as  a  possible  singular 
point  for  any  branch  off(z)  except  the  initial  branch. 

It  should  be  observed  that  y..  is  shown  to  be  a  potential  rather 
than  an  actual  singular  point.  When,  however,  it  is  such  a  point, 
the  character  of  the  point  depends  in  general  solely  upon  the  nature 
of  the  singularities  a.  and  /i^.  for  (1)  and  (2)  respectively.  This  fact 
was  noticed  by  Borel  and  demonstrated  in  the  following  manner. 
Let 

<    4-  <B  +  ct?  +  •  -  - 


110  THE  BOSTON  COLLOQUIUM. 


be  any  convergent  series  defining  a  function  ^(x)  which  is  regu 
lar  at  a..  Then  <f>2(x)  =  ^(x)  -f  <f>(x)  is  a  function  which  has  at 
a.  the  same  singularity  as  <f>(x).  The  combination  of  the  series 
for  <j)z(x)  and  for  -^(x)  by  Hadamard's  process  gives  the  function 


/2(x)  =  (a0  +  c0)60  +  (a.+c^x  +  K+ 
in  which 

/iOO  =  CA  +  Ci6i^  +  c^as2  -f  •  •  •  • 

Now  since  ^(JG)  is  regular  at  a.,  when  compounded  with  -^(a;) 
it  must  give  a  function  /j(&)  which  is  regular  at  */...  It  follows 
that  f2(x)  and  f(x)  have  the  same  singularity  at  y...  Thus  the 
nature  of  this  singular  point  is  not  altered  by  any  change  in  <f>(x) 
or  ^r(#)  which  does  not  affect  the  character  of  the  points  a.  and  /?.. 
It  depends  therefore  solely  upon  the  character  of  the  singularities 
compounded. 

Complications  arise  only  when  there  is  a  second  pair  of  singu 
larities  ak,  /3{  such  that 

y..  =  aft  =  ak/3r 

Clearly  the  resultant  singularity  is  then  dependent  upon  both 
pairs.  Their  effects  may  be  so  superimposed  as  to  create  an  ugly 
singularity,  or  they  may,  on  the  other  hand,  so  neutralize  each 
other  that  7^.  is  a  regular  point.  Very  simple  examples  of  the 
latter  occurrence  can  be  easily  given.  It  seems  probable  that 
when  7t..  is  but  once  a  product  of  an  a  by  a  £,  it  must  always  be 
a  singular  point,  but  this  has  not  yet  been  proved.  Its  demon 
stration  will  greatly  enhance  the  value  and  applicability  of  Hada- 
mard's  theorem,  for  then  it  can  be  stated  in  numerous  cases,  not 
what  the  singular  points  offlx)  may  be,  but  what  they  actually  are. 
A  detailed  study  of  the  nature  of  the  dependence  of  the  singu 
larity  7r  upon  a.  and  /3.  would  probably  be  both  interesting  and 
profitable.  Borel  examines  the  case  in  which  a.  and  /^  are  poles  of 
any  orders,  p  and  5-,  and  shows  that  ytj  is  then  a  pole  of  order 
p  _|_  q  —  l.  It  can,  furthermore,  be  easily  shown  that  whenever 
a.  is  a  pole  of  the  first  order,  7^  is  the  same  kind  of  singular  point 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    Ill 

as  ft..    For  suppose  that  we  put  a.  =  1 ,  which  may  be  done  without 
loss  of  generality.     The  principal  part  of  $(:?)  at  the  pole  a.  is  then 


and  the  composition  of  this  with  ^(x)  gives  for  the  corresponding 
component  of  f(x) 

-  Ai  (bQ  +  bfc  +  6,.r2  +  •••). 

Hence   the    singularities  ytj  and    ftj  differ    by  a    multiplicative 
constant. 

Only  one  other  general  fact  concerning  the  composition  of  sin 
gularities  seems  to  be  known.  Borel  proves  that  if  the  functions 
<t>(x)  and  "^(x)  are  one-  valued  at  a.  and  ft.  respectively,  f(x)  is 
also  one-valued  at  7i;.  Thus  when  two  one-valued  functions  are 
compounded,  the  resultant  function  is  also  one-valued.  But 
this  statement,  as  he  himself  points  out,  must  be  correctly  con 
strued  and  will  not  necessarily  hold  true  when  the  singular  points 
of  the  two  given  functions  are  not  sets  of  isolated  points  but  con 
dense  in  infinite  number  along  curves.  To  construct  an  example 
in  which  /(.?•)  in  not  one-valued,  Borel  makes  use  of  the  fact, 
now  so  well  known,  that  the  decision  whether  the  circle  of  con 
vergence  is  or  is  not  a  natural  boundary  of  a  given  series  depends 
upon  the  arguments  of  its  coefficients.  If,  for  instance,  we  take 
the  series 


which  has  a  radius  of  convergence  equal  to  1,  by  a  proper  choice 
of  the  arguments  0n  the  circle  of  convergence  can  be  made  a  natu 
ral  boundarv.  Put  now 


(6)  1/1  _  x  =  c0  +  ct  x  +  c2or  +  •  •  • , 

in  which  the  coefficients  are  necessarily  real.     Clearly  the  unit 
circle  will  be  a  natural  boundary  for 

(j>x  =  c  +  cx  -f  ce-x  -  +  . .  - . 


112  THE  BOSTON  COLLOQUIUM. 

and  for 

^(ic)  =  1  +  e-i0*x  4-  e~ie*x  2-f 

Yet  the  function  f(x)  which  is  derived  from  these  two  one- 
valued  functions  by  Hadamard's  process  is  the  two-  valued  func 
tion  (6)  which  exists  over  the  entire  plane  of  x. 

I  have  dwelt  at  some  length  upon  Hadamard?8  theorem  and  its 
consequences  because  of  their  evident  interest  and  importance.  It 
is  worthy  of  note  that  for  analytic  functions  defined  by  power 
series  the  first  great  advance  in  the  determination  of  the  singu 
larities  over  their  entire  domain  has  been  made  by  methods  that 
are  roughly  parallel  to  those  currently  employed  in  the  considera 
tion  of  their  convergence.  The  convergence  of  series  is  indeed 
too  difficult  a  question  to  be  settled  by  any  one  rule  or  by  any 
finite  set  of  rules,  but  the  methods  of  comparison  with  series  known 
to  be  convergent  have  been  found  to  be  not  only  most  efficient 
but  also  adequate  for  most  practical  purposes.  In  somewhat 
similar  fashion  Hadamard's  theorem  will  determine  the  singular 
points  of  numerous  functions  by  linking  them  with  other  series, 
of  which  the  singularities  are  known. 

One  of  the  simplest  applications  of  this  theorem  is  obtained  by 
compounding  a  given  series 

(7)  a0  4-  ap  4-  «2^2  +  v  • 

with  itself  once,  twice,  •  •  -,  to  m  times.  All  the  singularities  of 
the  resulting  series 

(8)  aj  4-  a[x  +  a\tf  +  .  .  -         (t  =  1,  2,  •  •  •  ,  m), 

except  possibly  x  =  0  and  x  =  oo,  are  included  among  the  points 
obtained  by  multiplying  i  affixes  of  the  singular  points  of  (7) 
among  themselves  in  all  possible  ways.  If  the  m  series  (8)  are 
multiplied  each  by  a  constant  k{  and  are  then  added,  a  new  series 


(9)  G(a0)+G(a>+q 

is  obtained,  in  which  G(u)  denotes  the  polynomial^  4-  •  •  •  4-  knu 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    113 

This  function  has  no  singular  points  other  than  those  which  are 
possible  for  the  m  series  from  which  it  was  derived.  "When  r 
different  series 

<70  4-  cijX  4-  a2x2  4-  •  •  • , 


r0  4-  ^  4-  /y^  4-  •  •  •  , 
are  used,  a  similar  conclusion  is  reached  for  the  series 


where  G  denotes  a  polynomial  in  which  the  constant  term  is  lacking. 
These  results  are  of  particular  interest  when   applied  to  the 
series 

(10)  1  +  x  4-  2x2  4-  •  .  •  4  nxn  4-  •  -  - 
and 

(11)  I  +  X  +  Y  +  -  •  +  ?+•". 

which  are  the  expansions  of  1  4-  a?/(l  —  a;)2  and  log  (1  -f  x).  Since 
these  functions  have  only  one  singular  point,  x  =  1,  in  the  finite 
plane,  the  only  possible  singularities  of 


are  x—  0,  1,  oo.* 

The  continued  repetition   of  the  above  process  for  combining 
series  leads  naturally  to  a  consideration  of  series  of  the  form 

(12)  2P(a>" 

in  which  a  convergent  power  series  P(u)  appears  in  place  of  the 

polynomial    G(u).      Various  theorems  concerning  cases  of  this 

*  Obviously  a  constant  term  can  be  included  now  in  the  polynomial  G(T?,  l/«). 


114  THE  BOSTON  COLLOQUIUM. 

series  have  been  given  recently  by  Lean*  Le  Roy^  Dexaint,$ 
Lindelof,§  Ford  \\  and  Faberfl  though  the  proof  of  some  of  these 
theorems  has  no  direct  relation  to  Hadamard's  theorem.  The 
importance  of  such  work  is,  however,  apparent,  inasmuch  as  nu 
merous  series  which  occur  in  analysis  can  be  put  into  the  form 
under  consideration,  as  for  example  2(sin  7r/n)xn. 

Three  cases  must  be  distinguished  according  as  the  radius  of 
convergence  of  the  initial  series  (7)  is  less  than,  equal  to,  or 
greater  than  1.  If  the  radius  is  less  than  1,  the  singular  point 
nearest  to  the  origin  has  a  modulus  less  than  1,  and  the  continued 
multiplication  of  the  affix  of  the  point  by  itself  gives  a  series  of 
points  which  approach  indefinitely  close  to  the  origin.  The  pre 
sumption,  therefore,  would  naturally  be  that  the  series  (12)  is  then 
•divergent,  but  this  is  very  far  from  being  always  true,  as  will  be 
seen  at  once  by  referring  to  the  series  2  (or"  sin  ajand  2(#w  cos  an) 
in  which  an  is  real.  The  applicability  of  Hadamard's  theorem 
consequently  ceases. 

The  case  in  which  the  radius  of  convergence  of  (7)  is  greater 
than  1  has  been  investigated  very  recently  by  Desaint.  In  this  case 
the  expected  theorem  is  obtained.  If,  namely,  P(u)  is  a  conver 
gent  series  without  a  constant  term,  2P(an)an  defines  a  function 
which  can  have  no  singular  points,  besides  x  =  0  and  x—  oo, 
than  those  which  result  from  the  multiplication  of  the  affixes  of 
the  singular  points  among  themselves  in  all  possible  ways  and  to 
any  number  of  times.**  Demint's  proof  is  based  upon  the  fact 
that  2P(an)sn,  after  the  omission  of  a  suitable  number  of  terms, 
can  be  expressed  in  the  form 

*  Journ.  de  Math.,  ser.  5,  vol.  5  (1899),  p.  365. 

fioc.  eft. 

J  Journ.  de  Math.,  ser.  5,  vol.  8  (1902),  p.  433. 

g  Ada  Societatis  Sdentiarum  Fennicce,  vol.  31  (1902). 

II  Journ.  de  Math.,  ser.  5,  vol.  9  (1903),  p.  223. 

^Math.  Ann.,  vol.  57  (1903),  p.  369. 

**  This  is  a  somewhat  sharper  statement  of  the  result  than  that  given  by  De- 
saint.  In  his  theorem  x  =  1  is  given  as  a  possible  singular  point,  but  this,  as 
appears  from  the  proof  to  be  given  here,  is  due  solely  to  the  admission  of  a  con 
stant  term  into  P(u).  He  also  fails  to  note  that  z  =  0  may  be  a  singular  point. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    115 

mm  •••/(<*) 


in  which  f(t)  is  the  function  defined  by  (7)  for  x  =  t,  c'  is  an  ap 
propriately  chosen  contour,  and  cn  denotes  the  nth  coefficient  of 

P(u)  =  cji  +  ctu*  +  .  •  .  , 

Although  his  proof  is  essentially  simple  in  character,  I  shall  give 
here  a  new  and  simpler  proof,  based  directly  upon  Hadamard's 
theorem. 
Place  first 

ffr)  =  aj  +  a[x  -f  ap  +  ....       (t  =  2,  3,  .  .  .), 
and  consider  the  expression 

'„/»  +  '.+  ,/.+  ,(*)+  — 

in  which  n  denotes  some  fixed  integer.  If  r>  1  denotes  the 
radius  of  convergence  of  the  fundamental  series  (7),  the  radius  of 
fj(x)  will  be  r'.  Describe  about  the  origin  a  circle  (r')  having  a 
radius  r'  <  ?>n.  If  a  sufficient  number  of  initial  terms  be  cut  off  in 
each  of  the  series, 


the  maximum  absolute  values  of  the  remainders  within  or  upon 
the  circle  (r')  can  be  made  as  small  as  is  desired.  Suppose  then 
that  after  m  terms  of  each  have  been  removed,  the  remainders 


do  not  exceed 

-n  n+1  f2n 

t    ,         t          ,  ,         t 

respectively,  in  which  e  is  some  small  positive  number.     Let  us 
now  substitute  in  Hadamard's  integral 


116  THE  BOSTON  COLLOQUIUM. 


/ 
J. 


any  two  of  the  functions  (13)  for  $  and 
Put  for  example 


and  choose  the  unit  circle  as  the  path  of  integration.  Then  if 
|«|=r',  the  absolute  values  of  the  arguments  of  the  series  <j>(zx) 
and  A/r(l/cc)  will  be  less  than  their  radii  of  convergence  since 
|  as  |  =  1  and  r  >  1.  The  conditions  for  the  existence  of  Hada- 
mard's  integral  are  therefore  fulfilled.  Since  also 


we  have 


But  by  Hadamard's  theorem  F(z)  —  r2n+i+i(z),  and  hence 

(14)  K(z)|<ej  (I»I^O, 

for  all  values  of  i  from  2n  to  4n  inclusive.  The  reasoning  can 
now  be  repeated  with  2n  in  place  of  n,  and  so  on ;  therefore  (14) 
is  true  for  all  values  of  i=  n. 

Thus  far  the  value  of  e  has  remained  arbitrary.  Let  its  value 
now  be  taken  less  than  the  radius  of  convergence  of  P  (u).  Then 
by  (14)  the  series 

(15)  c»'»  +  cn+1rn+1(x)  +  ... 

will  be  uniformly  convergent  in  (r').  Since,  furthermore,  all  the 
component  series  rn+i(i  =  0,  1,  2,  •  •  •)  are  likewise  so  convergent, 
by  a  fundamental  and  familiar  theorem  of  Weierstrass  *  the  terms 
of  the  collective  series  (15)  may  be  rearranged  into  an  ordinary 
series  in  ascending  powers  of  x.  But  this  rearrangement  gives 

*  Harkness  and  Morley's  Introduction  to  the  Theory  of  Analytic  Functions,  p.  134. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    117 


or  the  remainder  after  the  (m  —  l)th  power  of  x  in 
(16)        £  P(«.y  -  cj(x)  -  cJ2(x)  -----  c^f 


j=0 

Now  the  series  (15)  before  its  rearrangement  was  a  uniformly 
convergent  series  of  analytic  functions  and  defined  a  function 
which  was  analytic  within  (?•').  It  follows  that  (16)  is  also 
analytic  within  this  circle,  and  hence 


has  no  singularities  within  this  circle  except  those  of 

/,(*),  /*(*)>  •••,/.-,(»)• 

But  the  radius  of  (»•')  was  any  quantity  short  of  ?>n,  and  this  con 
clusion  therefore  holds  within  a  circle  having  its  center  in  the 
origin  and  a  radius  equal  to  rn.  By  increasing  n  indefinitely,  the 
theorem  of  Desaint  results.  It  is  evident  also  that  if  fi(z),  and 
therefore  />(#)>  represents  a  one-  valued  function,  2P(aJ#n  must 
also  be  such  a  function. 

There  remains  yet  for  consideration  the  third  class  of  cases  in 
which  the  radius  of  convergence  of  the  fundamental  series  is  1. 
If  upon  the  circle  of  convergence  there  is  any  singular  point  with 
an  incommensurable  argument,  the  continued  multiplication  of  its 
affix  by  itself  gives  a  set  of  points  everywhere  dense  upon  the 
circle  of  convergence.  It  is  therefore  to  be  expected  that  this 
circle  will  be,  in  general,  a  natural  boundary  for  2P(an)#n,  and 
accordingly  the  cases  which  will  be  of  chief  interest  will  be  those 
in  which  all  the  singular  points  upon  the  circle  have  commensur 
able  arguments.  A  simple  case  of  this  character  is  obtained 
when  either  (10)  or  (11)  is  chosen  as  the  generating  series.  If 
the  former  be  selected,  the  resulting  series  has  the  form  2P(?i)ajn. 
This  has  a  special  interest  inasmuch  as  its  study  has  proved  to 


118  THE  BOSTON  COLLOQUIUM. 

be  of  profit  both  for  the  theory  of  analytic  continuation  and  of 
divergent  series.  The  reason  becomes  apparent  when  the  state 
ment  is  made  that  it  is  possible  to  throw  any  Taylor's  series, 
2ano?w,  whether  convergent  or  divergent,  into  the  particular  form 
2P(w)jcn,  and  in  an  infinite  number  of  ways.  This  fact  follows 
as  a  corollary  of  a  very  general  theorem  of  Mittag-Leffler*  which, 
when  restricted  to  the  special  case  before  us,  establishes  the  exist 
ence  of  a  function  P(x),  which  is  holomorphic  over  the  entire 
finite  plane  and  assumes  the  pre-assigned  values  a0,  al9  «2,  •  •  •  in 
the  points  x  =  0,  1,  2,  •  •  -.  Consequently  the  character  of  the 
function  defined  by  2P(n)xn  is  made  to  depend  upon  the  behavior 
of  P(x)  as  x  approaches  oo. 

Inasmuch  as  2P(Vi)of  is  perfectly  general,  limitations  must  be 
imposed  upon  P(u)  in  any  attempt  to  extend  Hadamard's  theorem 
to  this  series.  But  whenever  the  theorem  is  applicable,  the  only 
possible  singularities  of  2P(n)o3"  are  x  =  0,  1,  oo.  Lean  f  estab 
lishes  the  correctness  of  this  result  when  P(u)  is  an  entire  function 
of  order  less  than  1,J  giving  also  a  more  general  theorem  §  concern 
ing  2P(an)of  of  which  this  is  a  special  case.  The  like  conclusion 
holds  concerning  the  singular  points  of  2P(l/n)of ,  provided  only 
that  P(x)  is  holomorphic  at  the  origin. || 

Very  recently  these  results  of  Lean  have  been  proved  more 
simply  by  Faber,  but  in  a  more  restricted  form,  an  artificial  cut 
being  drawn  from  x  =  1  to  x  =  oo  to  obtain  a  one  valued  func 
tion.  In  addition,  Faber  shows  that  if  for  any  prescribed  e  and 
for  a  sufficiently  large  r  the  inequality 

(17) 


*  Acta  Math.,  vol.  4  (1884),  p.  53,  theorem  D.  For  a  reference  to  this  theorem 
I  am  indebted  to  Professor  Osgood.  Theorem  2  of  Desaint's  memoir  (p.  438) 
is  in  contradiction  with  this,  but  his  proof  is  here  inadequate  since  rk  (p.  440)  has 
not  necessarily  a  lower  limit. 

•\Loc.  cit.,  p.  418. 

JHe  also  shows  that  2P(n)o;n  is  then  a  one-valued  function. 

%Loc.  cit.,  p.  417.  See  also  Butt,  de  la  Soc.  Math,  de  France,  vol.  26  (1898), 
p.  267. 

\\Loc.  cit.,  p.  418 ;  see  also  p.  407. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    119 

is  fulfilled,  the  point  x  =  1  must  be  an  essential  singularity,  and 
the  function  represented  by  ZP(n)xn  is  consequently  one-  valued.* 
Conversely,  if  f(x)  is  a  one-valued  function  which  has  only  one 
singular  point,  and  if  that  point  is  an  essential  singularity,  f(x) 
can  be  expressed  in  the  form  2P(>i).eu,  in  which  P(u)  is  an  entire 
function  satisfying  (17).  More  generally,  if  there  are  I  essential 
singularities  xlt  •  •  •,  xl  and  no  other  singular  points  in  the  finite 
plane,  the  coefficient  of  x*  must  be 


in  which  P^n),  •  -  >,  Pt(n)  are  entire  functions  of  the  nature  above 
specified  and  in  which  Km  \/a>'n  =  0.  This  converse  has  an  espe 
cial  interest  because  as  yet  few  theorems  have  been  discovered 
giving  the  necessary  form  of  the  coefficients  of  a  power  series  for 
an  analytic  function  with  prescribed  functional  properties. 

Other  theorems  concerning  ?LP(n)xn  have  recently  been  derived 
without  requiring  that  P(n)  shall  be  holomorphic  over  the  entire 
plane. 

As  a  sample  of  these  I  shall  cite  in  conclusion  the  following 
theorem  of  Linddof:  f 

If  P(n)  represents  a  function  fulfilling  the  following  conditions  : 

1.  P(z)    is    analytic    for   every  point  of  the    complex    plane 
z  =  r  -j-  it  for  which    r  =  0    (except  possibly   at  the  origin,  for 
which  P(z)  has  a  determinate  value). 

2.  A  number  e  being  arbitrarily  given,  it  is  possible  to  find 
another  number  R  such  that  by  putting  z  =  re1*  we  will  have  for 
r>=B 


*  Le  Roy  three  years  earlier  had  noted  this  conclusion  when  P(x)  is  an  entire 
function  whose  "apparent  order"  is  less  than  1  ;  loc.  cit,  p.  348,  footnote. 
Faber  does  not  seem  to  be  aware  of  Le  Roy's  statement.  The  difference  between 
the  two  statements  is  slight  but  becomes  important  in  formulating  the  new  and 
interesting  converse  which  Faber  adds. 

fioc.  cit.,  §  13. 


120  THE  BOSTON  COLLOQUIUM. 


then  the  principal  branch  of  the  function  2.P(n)3n  will  be  holo- 
morphic  throughout  the  complex  plane  excepting  possibly  on  the 
segment  (1,  -f  oo  )  of  the  real  axis.  Furthermore,  the  function 
approaches  0  as  a  limit  when  x  tends  toward  the  point  at  infinity 
along  any  ray  having  an  argument  between  0  and  2?r. 

LECTURE  IV.      On  /Series  of  Polynomials  and  of  Rational 

Fractions. 

In  the  last  two  lectures  I  have  spoken  of  the  use  of  integrals  for 
the  study  of  analytic  extension  and  of  divergent  series.  The  topic 
of  to-day's  lecture  is  the  representation  of  functions  by  means  of 
series  of  polynomials  and  of  rational  fractions.  This  subject  forms 
a  very  natural  transition  to  the  succeeding  lecture  upon  continued 
fractions,  since  an  algebraic  continued  fraction  is  in  reality  noth 
ing  but  a  series  of  rational  fractions  advantageously  chosen  for  the 
study  of  a  corresponding  function  which,  when  known,  is  com 
monly  given  in  the  form  of  a  power  series. 

The  literature  relating  directly  or  indirectly  to  series  of  poly 
nomials  and  of  rational  fractions  is  a  vast  one,  with  many  ramifi 
cations.  Thus  in  one  direction  there  are  various  researches  of 
importance  upon  the  non-uniform  convergence  of  series  of  contin 
uous  functions,  and  in  this  connection  I  may  refer  particularly  to 
the  recent  work  of  Osgood  and  Baire,  an  excellent  report  of  which 
is  contained  in  Schonflies*  Bericht  iiber  die  Mengenlehre.*  An 
other  part  of  the  field  comprises  numerous  memoirs  devoted  to 
special  series  of  polynomials  and  rational  fractions.  Quite  re 
cently  a  more  systematic  and  general  study  has  been  begun  by 
Borely  Mittag-Leffler  ,  and  others,  and  it  is  to  this  that  I  am  to 
call  your  especial  attention. 

Two  very  familiar  facts,  both  discovered  by  Weierstrass,  may 
be  said  to  be  the  origin  of  this  study.  I  refer,  of  course,  to  the 
theorem  that  any  function  which  is  continuous  in  a  given  finite 
interval  of  the  real  axis  can  be  expressed  in  that  interval  as  an 

*  Jahresbericht  der  deutschen  Mathematilcer-  Vereiniyung,  vol.  8,  pp.  224-241. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    121 

absolutely  and  uniformly  convergent  series  of  polynomials,*  and, 
secondly,  to  the  possibility  that  a  single  series  of  rational  fractions 
may  represent  two  or  more  distinct  analytic  functions  in  different 
portions  of  its  domain  of  convergence.  A  notable  advance  upon 
the  theorem  first  mentioned  was  made  by  Runge^  in  1884,  who 
proved  that  any  one-valued  analytic  function  throughout  the  do 
main  of  its  existence  can  be  represented  by  a  series  of  rational  ; 
functions;  furthermore,  this  domain  may  be  of  any  shape  what-  ! 
soever,  provided  only  it  forms  a  two-dimensional  continuum. 
Range's  proof  of  these  important  results  is  not  only  worthy  of 
careful  study,  but  contains  also  certain  conclusions  which  were  an 
nounced  again  by  Painleve%  in  1898,  though  without  proof. 
The  conclusions  reached  were  as  follows  : 

Let  D  be  a  domain  consisting  of  any  number  of  separate  pieces 
of  the  complex  plane,  in  each  of  which  we  will  suppose  an  analytic 
function  to  be  defined.  The  functions  thus  defined  can  be,  at 
pleasure,  either  distinct  functions  or  parts  of  one  or  more  func 
tions.  In  any  case  a  series  of  rational  functions  can  be  formed 
which  will  converge  absolutely  and  uniformly  in  any  region 
lying  in  the  interior  of  D  and  represent  in  each  separate  piece 
the  prescribed  function.  Furthermore,  this  representation  can 
be  made  in  an  infinite  number  of  ways.  Let  the  ensemble  of 
the  points  excluded  from  D  be  represented  by  E.  When  E  con 
sists  of  a  single  connected  continuum  of  any  sort,  whether  linear 
or  areal,  any  point  a  of  E  can  be  arbitrarily  selected,  and  the 
function  can  be  expanded  into  the  series 


"  Ueber  die  analytische  Darstellbarkeit  sogenannter  willkiirlicher  Functionen 
einer  reellen  Veriinderlichen ' ' ;  Berliner  Sitzungsberichte,  1885,  p.  633  or  Werke, 
vol.  3,  p.  1.  Simple  proofs  of  the  theorem  have  been  given  by  Lebesque,  Butt, 
des  Sciences  Math.,  ser.  2,  vol.  22  (1898),  p.  278,  and  by  Mittag-Leffler,  Rendicvnti 
di  Palermo,  vol.  14  (1900),  p.  217,  with  an  extension  to  functions  of  two  variables. 
In  this  connection  see  Painleve"snote  in  the  Compt.  Rend.,  vol.  126  (1898),  p.  459. 

•Mcta  Math.,  vol.  6,  p.  229. 

J  Compt.  Rend.,  vol.  126,  pp.  201  and  318. 


122  THE  BOSTON  COLLOQUIUM. 

in  which  Gn  [1  /(x  —  «)]  denotes  a  polynomial  in  1  /  (x  —  a).  If,  in 
particular,  the  continuum  E  contains  the  point  x—  oo,  an  ordinary 
series  of  polynomials,  26rn(V),  can  be  employed.  When  ^con 
sists  of  a  finite  number  of  separate  pieces  (or  isolated  points),  the 
expansion  can  be  put  under  the  form 


££(D  (—L-\  +  jr 

»   \x  -  aj       ~ 


in  which  aA,  •  •  •  ,  a  are  points  arbitrarily  chosen  in  the  separate 
pieces. 

In  the  familiar  case  in  which  only  a  single  analytic  function 

% 

(1)  aQ  +  aj(x  —  a)+  a20  —  a)2  + 

is  given,  it  is  natural  to  seek  a  series  of  polynomials  having  the 
greatest  possible  domain  of  convergence.  Unless  the  function  is 
one-  valued,  the  most  convenient  domain  is  in  general  the  star  of 
Mittag-Leffler.  This  is  constructed  for  the  series  (1)  by  first  marking 
on  each  ray  which  terminates  in  a  the  nearest  singular  point  and  then 
obliterating  the  portion  of  the  ray  beyond  this  point.  The  region 
which  remains  when  this  has  been  done  is  a  star  having  a  for  its 
center.  Mittag-Leffler  *  shows  that  within  the  star  the  given  ana 
lytic  function  can  be  represented  by  a  series  of  polynomials  in 
which  the  coefficients  of  the  polynomials  depend  only  upon  the 
value  of  the  function  and  its  derivatives  at  a,  t  or,  in  other  words, 
upon  the  coefficients  of  (1).  If,  in  short,  we  put 


and 


.  n 

Ai=0  A2=0        A,v=0  •          *  '  U 


*  Ada  Math.,  vol/23  (1899),  p.  43  ;  vol.  24,  pp.  183,  205  ;  vol.  26,  p.  353.  A 
good  summary  is  found  in  the  Proc.  of  the  London  Math.  Soc.,  vol.  32  (1900),  pp. 
72-78. 

t  In  this  respect  his  work  is  superior  to  that  of  Runge  and  others.  Range, 
for  example,  presupposes  a  knowledge  of  the  function  at  an  infinite  number  of 
points. 


DIVERGENT  SERIES  AXD  CONTINUED  FRACTIONS.     123 

then  5^  ^rn(#)  is  a  series  which  converges  uniformly  in  any  region 

»=a 
lying,  with  its  boundary,  entirely  in  the  interior  of  the  star.     The 

series  may  also .  converge  outside  the  star.  Borel*  has  shown, 
furthermore,  that  the  series  of  Mittay-Leffler  is  not  the  only  possi 
ble  one,  but  there  is  an  infinity  of  polynomial  series  sharing  the 
same  property  within  the  star. 

It  will  be  noticed  that  the  construction  of  the  series  of  Mittag- 
Leffler  is  in  no  wise  dependent  upon  the  convergence  of  the  initial 
.  power  series.  In  certain  cases,  at  least,  the  polynomial  series  con 
verges  when  the  given  series  (1)  is  itself  divergent.  It  is  natural 
therefore  to  look  for  a  theory  of  divergent  series  based  upon  con 
vergent  series  of  polynomials.  As  yet,  however,  no  such  theory 
has  been  invented.  One  of  the  chief  difficulties  in  the  way  is  that 
the  polynomial  series  do  not  afford  a  unique  mode  of  representing 
an  analytic  function.  Xow  the  difference  between  any  two  series 
of  polynomials  for  the  same  function  in  an  assigned  area  is  a  third 
series  which  vanishes  at  every  point  of  the  area,  though  the  sep 
arate  terms  do  not.  This  is  a  decidedly  awkward  point,  and 
occasions  difficulty  in  proving  or  disproving  the  identity  of  two 
functions  expressed  by  polynomial  series.  It  is  true,  indeed,  that 
this  difficulty  will  scarcely  present  itself  when  we  start  with  a  con 
vergent  power  series  which  is  to  be  continued  analytically,  the 
polynomial  series  then  giving  continuations  of  a  common  function. 
But  when  the  series  (1)  is  divergent  and  there  is  no  known  func 
tion  which  it  represents,  it  is  an  open  question  whether  the  differ 
ent  series  of  polynomials  which  are  obtained  from  (1)  by  applica 
tion  of  diverse  laws  will  furnish  the  same  or  different  functions. 
If  different  functions,  is  there  any  ground  for  preferring  one  series 
of  polynomials  to  another  ? 

Up  to  the  present  time  two  essentially  different  principles  seem 
to  have  been  followed  in  the  formation  of  series  of  polynomials. 
In  the  work  of  Rung?,  Borel,  Painlei-e  and  Mittag-Leffler  the  co 
efficients  in  the  polynomials  vary  with  the  character  of  the  ana- 

*Ann.  de  r EC.  Sor.,  ser.  2,  vol.  16  (1899),  p.  132,  or  Les  Series  divergentes, 
p.  171. 


124  THE  BOSTON  COLLOQUIUM. 

lytic  function  to  be  represented  ;  for  example,  in  the  polynomials 
of  Mittag-Leffler  they  are  functions  of  the  coefficients  of  the  given 
element,  ^anxn.  By  appropriately  choosing  the  coefficients  of  the 
polynomials  these  writers  obtain  a  very  large  region  of  conver 
gence  and  at  the  same  time  are  able  to  greatly  vary  its  shape.  On 
the  other  hand,  the  series  which  are  met  in  the  practical  branches 
of  mathematics  —  for  instance,  in  the  theory  of  zonal  harmonics  — 
have  the  form 

(2)  c0<yo(aO  +  c,  <?,(*)  +  otGJx)  +  ••• 

in  which  the  polynomials  Crn(x)  are  entirely  independent  of  the 
function  represented,  while  the  c{  vary.  The  polynomials  them 
selves  are  selected  according  to  the  shape  of  the  region  of  con 
vergence.  Thus  if  the  region  is  a  circle,  we  may  put 


and  we  have  then  the  ordinary  Taylor's  series.  Or  if  it  be  an 
ellipse  having  the  foci  ±  1,  we  may  take  for  our  polynomials 
either  the  successive  zonal  harmonics  or  a  second  succession  of 
polynomials  (also  called  Legendre's  polynomials)  which  are  con 
nected  by  the  recurrent  relation  : 

(3)       G^x)  -  2x(2n  +  3)  Gn+i(x)  +  4(n  +  !)«(?»  =  0. 

In  a  recent  number  of  the  Mathematische  Annalen  (July,  1903) 
Faber  has  considered  this  second  class  of  polynomials  from  a  some 
what  general  point  of  view  and  has  demonstrated  that  any  function 
which  is  holomorphic  within  a  closed  branch  of  a  single  analytic 
curve,  as  for  example  an  ellipse  or  a  lemniscate  of  one  oval, 
can  be  expressed  by  a  series  of  the  form  (2).  The  properties 
of  his  series  are  similar  to  those  of  Taylor's  series.  In  the 
case  of  the  latter,  to  ascertain  whether  2anccn  converges  in  the 
interior  of  a  circle  having  its  center  in  the  origin  and  a 
radius  R,  we  have  only  to  determine  the  maximum  modulus  of 
a  point  of  condensation  of  the  set  of  points  v/aB  (n  =  1,  2,  3,  •  •  •). 
If  it  is  exactly  equal  to  l/R,  the  circle  (R)  is  the  circle  of  con- 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     125 

vergence,  and  there  is  at  least  one  singularity  upon  its  circumfer 
ence.  If,  on  the  other  hand,  it  is  greater  or  less  than  1;J2,  the 
series  will  have  a  smaller  or  a  larger  circle  of  convergence.  So  also 
to  the  given  branch  of  the  analytic  curve  there  corresponds  a 
certain  critical  value.  When  this  is  exactly  equal  to  the  upper 
limit  of  ]/cn|  in  Faber's  series,  the  given  analytic  branch  is  the 
curve  of  convergence.  At  every  point  within,  the  series  converges, 
while  it  diverges  at  every  exterior  point,  and  upon  the  curve  there 
must  lie  at  least  one  singular  point  of  the  function  defined  by  (2). 
If,  however,  the  upper  limit  is  greater  or  less  than  the  critical 
value,  we  consider  a  certain  series  of  simple,  closed  analytic  curves, 
(as  for  example  a  series  of  confocal  ellipses),  among  which  the  given 
analytic  branch  must,  of  course,  be  included.  The  curve  of  con 
vergence  is  then  fixed  by  the  reciprocal  of  the  upper  limit  of 
|  yxcn  |  provided  this  limit  is  not  too  large.  Moreover,  as  in  the 
case  of  Taylor's  series,  the  function  cannot  vanish  identically  un 
less  every  cn  ==  0,  and  in  consequence  the  series  vanishes  identi 
cally.  It  is  therefore  impossible  that  the  same  function  shall  be 
represented  by  two  different  series  of  the  given  form. 

In  view  of  the  last  mentioned  fact  it  might  be  of  especial  inter 
est  to  apply  this  class  of  polynomial  series  to  the  study  of  diver 
gent  series. 

In  the  most  familiar  and  useful  polynomial  series  the  successive 
polynomials  are  connected  by  a  linear  law  of  recurrence, 

(4)  *,<?„+.,(*)  +  *,  <?„+,„-,«  +  •  •  •  +  W*)  =  0, 

in  which  the  coefficients  k.  are  polynomials  in  x  and  n.     Thus  the 
zonal  harmonics  have  as  their  law  of  recurrence 


Many  series  of  this  nature  are  also  included  in  the  class  con 
sidered  by  Faber.  The  form  of  the  region  of  convergence  has 
been  determined  by  Poincare  *  upon  the  hypothesis  that  equation 

*Amer.  Journ.  qf  Math.,  vol.  7  (1885),  p.  243. 


126  THE  BOSTON  COLLOQUIUM. 

(4)  has  a  limiting  form  for  n  =  oo.     Let  the  equation  be  first 
divided  through  by  kQ)  and  then  denote  the  limits  of  the  successive 
coefficients  for  n  =  oo  by  ^(x),  k2(x),  •  •      &m(^)«     Construct  next 
the  auxiliary  equation 

(5)  zm  +  \(x)zm-1  +  k2(x)zn-z  +  •  -  •  +  km(x)  =  0. 

Except  for  particular  values  of  x  there  will  be  one  root  of  this 
equation  which  has  a  larger  modulus  than  any  other.  Let  r(x) 
be  that  root.  Poinoar^*  shows  that  with  increasing  n  the  ratio 
Grn(%)l  Grn-\(x)  wiU  approach,  in  general,  this  root  as  its  limit. 
The  region  of  convergence  is  therefore  confined  by  a  curve  of  the 
form  C  —  \r(x)\,  and  the  value  of  C  for  the  series  (2)  is  to  be 
taken  equal  to  the  radius  of  convergence  of  2cn?/n.f 

By  way  of  illustration  let  us  take  the  series  2cn6rn(o;)  in  which 
the  polynomial  obeys  the  law 

*More  specifically,  Poincare"  proves  that  if  no  two  roots  of  (5)  are  of  equal 
modulus,  QH(x)}Gn—i(x)  has  always  a  limit,  and  this  limit  is  equal  to  some  root  of 
(5),  usually  the  one  of  greatest  modulus. 

f  Poincare  has  given  no  proof  that  the  series  (2)  will  converge  at  those  points 
within  the  curve  |  r(x)  \  =  (7,  for  which  there  are  two  or  more  distinct  roots  of 
(5)  having  a  common  modulus  greater  than  the  moduli  of  the  remaining  roots. 
Thus  in  the  example  which  is  quoted  below  (p.  127),  these  are  the  points  of  the 
real  axis  which  are  included  between  -f-  1  and  —  1.  This  gap  in  Poincare'  s  theory 
can  be  filled  in  by  the  following  theorem  which  I  have  given  in  the  Transactions 
of  the  Amer.  Math.  Soc.,  vol.  1  (1900),  p.  298:  If  the  coefficients  in  the  series 
2Anyn  are  connected  by  a  recurrent  relation  having  the  limiting  form 

•"•  n  ~\~  K\An  —  i  ~T~  •  •  •  ~T"  Km  An  —  m  ——  Oj 

the  series  will  converge  at  the  worst  within  a  circle  whose  radius  is  the  recipro 
cal  of  the  greatest  modulus  of  any  root  of  the  auxiliary  equation 


Denote  this  maximum  by  7*,  irrespective  of  the  number  of  roots  having  this  maximum 

modulus.     Then 

\An\<M(r+e)»  (w=l,  2,  ...  ). 

Hence  if  C  is  the  radius  of  convergence  of  2c,$n,  the  series  2cnAn  will  converge 
when  C^>r.  Suppose  now  that  An  depends  upon  x  and  put  An  =  Gn(x).  It 
follows  then  from  my  theorem  that  2cnGn(x)  will  always  converge  when  tf>  r. 
But  this  is  what  was  to  be  proved. 

At  the  time  of  the  publication  of  my  work  I  was  not  aware  of  Poincare'  s  article, 
and  I  therefore  failed  to  point  out  the  relation  of  the  two  memoirs. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    127 

For  n  =oc  the  limiting  form  of  this  equation  is 


or  the  same  as  the  limiting  form  for  the  zonal  harmonic.     The 
auxiliary  equation  is 

S2  _  2xz  +  1  =  0, 
of  which  the  roots  are 


The  curves  |  x  ±  i  x*2  —  1  j  =  C  are  easily  seen  to  be  ellipses 
having  the  foci  =fc  1.  Hence  if  R  is  the  radius  of  convergence 
of  2cn#",  the  region  of  convergence  of  (2)  is  the  interior  of  an 
ellipse, 

|  x  =fc  vV^l  |  =  R. 

Poincare  also  examines  such  exceptional  cases  as  that  which  is 
specified  by  relation  (3),  which  has  no  proper  limiting  form.  But 
upon  this  work  we  can  not  longer  dwell.  I  wish,  however,  to 
emphasize  its  fundamental  character,  inasmuch  as  many  previous, 
and  even  subsequent  conclusions  concerning  the  convergence  of 
series  of  the  form  (2)  are  comprised  in  Poincar&s  result. 

Somewhat  earlier  in  the  lecture  I  set  forth  the  arbitrary  charac 
ter  of  the  function  which  could  be  represented  by  series  of  poly 
nomials  and  rational  fractions.  We  have  seen  also  how  this  arbi 
trary  element  was  entirely  eradicated  by  confining  ourselves  to 
polynomials  which  obey  a  linear  law  of  recurrence.  In  the  remain 
der  of  this  lecture  I  wish  to  develop  the  consequences  of  restrict 
ing  a  series  of  rational  fractions  in  the  manner  supposed  by  Borel  in 
his  thesis  *  and  its  recent  continuation  in  the  Ada  Mathematical 
Borel  seeks  to  so  restrict  a  series  of  rational  fractions,  2Pfi(#)/JRn(.r), 
as  to  ensure  a  connection  between  the  position  of  the  poles  of  its  sep 
arate  terms  and  the  position  of  the  singular  points  of  the  function 
which  the  series  collectively  represents.  On  this  account  he  assigns 

*Ann.  de  P  EC.  JVor.,  ser.  3,  vol.  12  (1895),  p.  1. 
fVol.  24  (1900),  p.  309. 


128  THE  BOSTON  COLLOQUIUM. 

au  upper  limit  to  the  degrees  of  Pn  (x)  and  Rn(x).  But  this  is  not 
enough,  and  he  proceeds  therefore  to  limit  the  magnitude  of  the  co 
efficients  in  the  numerators.  On  the  other  hand,  he  allows  any  dis 
tribution  whatsoever  for  the  roots  of  the  denominators,  thus  leaving 
himself  at  liberty  to  vary  greatly  the  nature  of  the  function  rep 
resented. 

In  his  thesis  he  develops  the  case 


(6) 


which  had  been  previously  considered  by  Poincare  *  and  Goursat.-f 
To  avoid  semi-convergent  series  or,  in  other  words,  functions,  of 
which  the  character  depends  not  merely  upon  the  position  of  the 
poles  an  and  the  values  of  An  but  also  upon  the  order  of  summation, 
the  condition  is  imposed  that  ^An  shall  be  absolutely  convergent. 
Then  if  there  is  any  area  of  the  z  plane  which  contains  no  poles, 
the  series  (6)  must  converge  within  this  region.  Since  further 
more  it  is  uniformly  convergent  in  any  interior  sub-region,  it 
defines  an  analytic  function  within  the  area.  There  may  be 
several  such  areas  separated  by  lines  or  regions  in  which  the  poles 
are  everywhere  dense.  This  is  precisely  the  case  to  be  considered 
now. 

To  simplify  matters,  let  us  suppose  that  the  poles  are  every 
where  dense  along  certain  closed  curves  of  ordinary  character, 
but  nowhere  inside  the  curves.  Poincare  and  Goursat  show  that 
each  curve  is  a  natural  boundary  for  the  analytic  function  <f>(z) 
defined  by  (6)  in  its  interior.  BorePs  proof  is  as  follows.  De 
note  the  component  of  (6)  which  corresponds  to  an  by 

" 


and  the  remaining  part  by 


*  Acta  Societatis  Fennicce,  vol.  12  (1883),  p.  341,  and  Amer.  Journ.  of  Math. 
vol.  14  (1892),  p.  201. 

f  Compt.  Rend.,  vol.  94  (1882),  p.  715. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     129 

A'  *  A" 

^(2) = £  c^r< + ,£,  F^fe* 

It  is  evident  that  if  «rt  lies  within  any  one  of  the  curves  considered, 
an  is  a  pole  of  (f>(z).  Now  when  these  interior  poles  condense  in  in 
finite  number  in  the  vicinity  of  any  point  of  the  curve,  it  must, 
of  course,  be  a  singularity  of  <f>(z).  Consider  next  any  one  of  the 
points  an  which  lies  upon  the  boundary  but  is  not  a  point  of  con 
densation  of  the  interior  poles,  and  let  z  approach  this  point  along 
the  normal.  Describe  a  circle  upon  the  line  z  —  an  as  diameter. 
If  2  is  sufficiently  near  to  anJ  the  circle  will  exclude  every  one  of 
the  points  at.,  excepting  an  which  lies  upon  its  boundary.  Since 
also  2An  is  absolutely  convergent,  by  increasing  r  the  second 
component  of  <£2(z)  may  be  made  less  in  absolute  value  than 
e/\z  —  an  m,  in  which  e  is  an  arbitrarily  small  prescribed  quantity. 
If,  them,  H  denotes  the  maximum  of  the  first  component  of  </>2(2) 
as  z  now  moves  up  to  an,  we  have 


i  n 

Consequently, 

lim  <t>(z)  •  (z  -  an)»  =  lim  <f>,(z)  •  (z  -  aa)m  +  lim  <f>.2(z)  -  (z  -  an)m=Bn. 

2=1n 

This  shows  that  ^(2)  |  increases  indefinitely  when  z  approaches 
any  pole  an  of  the  ??ith  order  along  a  normal,  and  removes  the  pos 
sibility  that  the  poles,  because  they  are  infinitely  thick  upon  the 
curve,  may  so  neutralize  one  another  that  the  function  can  be  car 
ried  analytically  across  the  curve  at  an.  As,  moreover,  we  sup 
pose  the  points  an  of  order  m  to  be  everywhere  dense  upon  the 
curve,  it  must  be  a  natural  boundary. 

It  is  apparent  now  that  the  expression  (6)  continues  the  initial 
function  <f>(z)  across  a  natural  boundary  into  other  regions  where 
it  defines  in  similar  manner  other  analytic  functions  with  natural 
boundaries.  But,  it  may  be  asked,  is  there  any  proper  sense  in 
which  these  analytic  functions  may  be  regarded  as  a  continuation 
of  one  another?  Just  here  Borel  steps  in  and,  after  imposing 


130  THE  BOSTON  COLLOQUIUM. 

further  conditions,  shows  that  when  the  function  defined  by  (6) 
within  some  one  of  the  curves  is  zero,  the  functions  defined  within 
the  other  curves  must  also  vanish.*  Take  m  =  1,  so  that 

(7)  <£0)  =  2  -A-  . 

v  '  z—  a 

n 

By  a  linear  transformation 

az  4-  b 


cz  -f  d 


any  interior  point  of  one  curve  may  be  taken  as  the  origin  and 
any  interior  point  of  a  second  curve  may  be  transformed  simul 
taneously  into  the  point  at  infinity  without  changing  the  character 
of  the  series  to  be  investigated.  Now  at  the  origin  the  successive 
coefficients  in  the  expansion  of  (f>(z)  into  a  Taylor's  series  are  the 
negative  of 

(8)  2^,      24s,      2-8",  •" 

«n  <  < 

while  those  in  the  expansion  for  z  =  oo  are 

(9)  24,     2,1  .«_     24X,  .... 

Borel  proves  that  when 

lim  y  '  An  —  0, 

n=co 

the  coefficients  (9)  must  vanish  if  those  given  in  (8)  do.  Any  one 
of  the  analytic  functions  under  discussion  is  therefore  completely 
determined  by  any  other,  the  expression  (7)  being  the  intermediary 
by  which  we  pass  from  one  to  the  other. 

So  far  as  yet  appears,  this  method  of  continuing  an  analytic 
function  across  a  natural  boundary  is  of  very  limited  applicability. 
Its  significance  has  been  made  clearer  by  Borel'  s  later  memoir  in 
the  Ada  Mathematica.  Here  the  rational  fractions  are  of  a  less  highly 
specialized  character,  but  the  essential  nature  of  the  investigation 
can  still  be  exhibited  without  abandoning  the  expression  (6).  Let 
I  AJ  <  1C+S  where  un  denotes  the  ?ith  term  of  a  convergent  series 

*  Cf.  pp.  32-33  of  his  thesis  or  pp.  94-98  of  his  Theorie  des  fonctions. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     131 

of  positive  numbers.  AVe  shall  suppose  that  the  poles  of  the  terms  of 
(6)  are  everywhere  dense  over  a  large  portion  of  the  plane,  leaving, 
however,  at  least  one  area  free  from  poles,  so  that  there  shall  be  an 
analytic  function  to  continue,  though  even  this  is  not  necessary. 
Borel  proves  that  parallel  to  any  assigned  direction  there  will  be 
an  infinity  of  straight  lines,  everywhere  dense  throughout  the 
plane,  along  which  the  series  (6)  will  converge  absolutely  and 
uniformly.  The  "function  defined  along  these  lines  is  therefore  a 
continuous  one. 

The  proof  of  this  result  is  short  and  simple.  Describe  about 
the  poles  a  as  centers  circles  which  have  successively  the  radii 
un(n  =  1,  2,  •  •  •).  If  there  is  any  point  which  lies  outside  all  of 
these  circles,  the  series  (6)  must  there  converge,  since  at  such  a 
point  the  absolute  value  of  the  nth  term  is 


that  is,  less  than  the  nth  term  of  a  convergent  series  of  positive 
numbers.  But  are  there  points  outside  of  all  the  circles  ?  To 
settle  this  question,  take  any  straight  line  perpendicular  to  the 
assigned  direction  and  project  orthogonally  all  the  circles  upon  the 
line.  The  total  sum  of  all  the  projections,  22  UB,  will  be  conver 
gent.  Moreover,  by  cutting  off  a  sufficient  number  of  terms  at 
the  beginning  of  (6),  the  sum  of  the  projections  may  be  made  less 
than  any  assigned  segment  ab  of  the  line.  Let  JV  terms  be  cut 
off  for  this  purpose.  Take  any  point  c  of  the  segment  which  does 
not  lie  upon  the  projection  of  any  circle  nor  coincide  with  the 
projection  of  one  of  the  first  ^V  poles  of  (6).  At  c  erect  a  perpen 
dicular  to  ab.  This  will  be  a  line  parallel  to  the  assigned  direc 
tion  which  throughout  its  entire  extent  lies  without  all  the  circles, 
excepting  possibly  the  first  A".  Hence  the  series  (6)  will  con 
verge  absolutely  and  uniformly  along  the  line,  even  though  the 
line  lie  infiuitesimally  close  to  some  set  of  poles  in  the  system. 
Lastly,  because  ab  was  an  interval  of  arbitrary  length,  these  lines 
of  convergence  must  be  everywhere  dense  throughout  the  plane, 
obviously  forming  a  non-enumerable  aggregate. 


132  THE  BOSTON  COLLOQUIUM. 

Since  the  series  is  uniformly  convergent,  it  can  be  integrated 
term  by  term.  Clearly  also  the  numerators  A.  in  (6)  can  be  so 
conditioned  that  the  term-by-term  derivative  of  (6)  shall  be 
uniformly  convergent.  Then  the  derivative  of  (f>(z)  is  coincident 
with  the  derivative  of  the  series.  It  is  even  possible  to  so  choose 
the  Ai  that  the  series  will  be  unlimitedly  differ  en  tiable. 

I  may  add  that  in  any  region  of  the  plane  there  will  be  an 
infinite  or,  more  specifically,  a  non-enumerable  set  of  points, 
through  each  of  which  passes  an  infinite  number  of  lines  of  con 
vergence.  If  a  closed  curve  is  given  it  will  be  possible  to 
approximate  as  closely  as  desired  to  this  curve  by  a  rectilinear 
polygon,  along  whose  entire  length  the  series  converges  and  defines 
a  continuous  function.  Integration  around  such  a  polygon  gives 
for  the  value  of  the  integral  the  product  of  2i?r  into  the  sum  of 
the  residues  of  those  fractions  whose  poles  lie  in  the  interior  of 
the  polygon.  Finally,  if  we  take  for  axes  of  x  and  y  two  perpen 
dicular  lines  of  continuity  of  </>(z),  all  the  lines  of  uniform  continuity 
which  meet  at  their  intersection  will  give  a  common  value  for  <j>'(z\ 
and  the  real  and  imaginary  parts  of  cf>(z)  will  satisfy  Laplace's 
equation : 

d2u      dzu 


Thus  we  have  in  $(%)  a  species  of  quasi-monogenic  function. 
One  question  Borel  has  as  yet  found  himself  unable  to  resolve. 
If  <j)(z)  =  0  along  a  finite  portion  of  any  line,  will  the  series  in 
consequence  vanish  identically  ?  If  this  question  be  answered  in 
the  affirmative,  the  analogy  with  an  ordinary  analytic  function 
will  be  still  more  complete. 

Let  us  now  return  to  the  case  in  which  two  or  more  functions 
with  natural  boundaries  are  defined  by  (7).  The  lines  of  con 
tinuity  just  described  form  an  infinitely  thick  mesh-work  along 
which  (f>(z)  can  be  carried  continuously  from  the  one  analytic 
function  into  the  others.  Suppose  again  that  the  origin  is  not  a 
point  of  condensation  of  the  poles  an  so  that  <£(»)  can  be  expanded 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    133 

at  the  origin  into  a  Maclaurin's  series  Sc^z1'.  Now  if  a  ray  is 
drawn  from  the  origin  through  the  pole  an  and  the  portion  of  the 
ray  between  an  and  oo  is  retained  as  a  cut,  the  ??*th  term  of  (7)  can 
be  expanded  into  a  series  of  polynomials 


which  converges  over  the  plane  so  cut.     The  series  (7)  can  there 
fore  be  resolved  into  a  double  series 


and  this  expression  will  be  valid  on  an  infinity  of  rays  from  the  origin 
which  do  not  pass  through  any  of  the  poles.  Since,  moreover,  the 
poles  are  an  enumerable  set  of  points,  these  rays  will  be  infinitely 
dense  between  any  two  arguments  which  may  be  taken.  By  fur 
ther  conditioning  the  An,  Borel  is  able  to  rearrange  the  terms  of 
the  double  series  so  as  to  form  a  series  of  polynomials  ^  Qn(z\ 

n 

in  which 


and  in  this  way  he  obtains  a  series  of  polynomials  which  is  con 
vergent  on  a  dense  set  of  rays  through  the  origin. 

It  also  appears  that  the  polynomial  series  2  Qn(z)  can  be  formed 
directly  from  Sc^'  without  the  intervention  of  (7).  When,  there 
fore  a  Maclaurin's  series  is  given  which  corresponds  to  such  an 
expression  (7)  as  is  now  under  discussion,  the  continuation  of  the 
function  can  be  made  along  the  above  set  of  rays.  Now  the  rays 
cut  any  curve  upon  which  either  (7)  or  ^Qn(z)  defines  a  continuous 
function  in  a  set  of  points  everywhere  dense.  The  value  of  the 
function  along  the  entire  curve  therefore  depends  only  upon  the 
coefficients  c. ;  /.  e.,  upon  the  value  of  the  function  and  its  deriva 
tives  at  the  origin.  It  is  shown,  moreover,  that  any  point  of  the 
plane  which  is  not  a  point  of  condensation  of  the  poles  an  may 


134  THE  BOSTON  COLLOQUIUM. 

be  converted  by  transformation  of  axes  into  such  an  origin. 
Finally,  Borel  gives  a  case  in  which  the  poles  may  be  everywhere 
dense  over  the  entire  plane,  so  that  the  function  defined  by  (7)  is 
nowhere  analytic,  and  yet  its  value  is  determined  along  the  lines 
of  continuity  by  the  value  of  the  function  and  its  derivatives  at 
the  origin.  Here  then  is  a  class  of  non-analytic  functions  sharing  a 
most  fundamental  property  in  common  with  the  analytic  functions! 
Is  it  not  then  possible,  as  Borel  surmises,  that  there  is  a  wider 
theory  of  functions,  similar  in  its  outlines  to  the  theory  of  ana 
lytic  functions  and  embracing  this  as  a  special  case?  If  so,  the  con 
ceptions  of  Weierstrass  and  of  Meray  are  capable  of  generalization. 


PART  II.     ON  ALGEBRAIC  CONTINUED  FRACTIONS. 

LECTURE  5.     Pad&s  Table  of  Approximates  and  its 
Applications. 

Both  historically  and  prospectively  one  of  the  most  suggestive 
and  important  methods  of  investigating  divergent  power  series  is 
by  the  instrumentality  of  algebraic  continued  fractions.  It  is  for 
this  reason  that  I  have  ventured  to  combine  in  a  single  course  of 
lectures  two  subjects  apparently  so  unrelated  as  divergent  series 
and  continued  fractions.  I  shall  not,  however,  confine  myself  to 
the  consideration  of  the  latter  subject  solely  with  reference  to  the 
theory  of  divergent  series.  It  is  rather  my  purpose  to  give  some 
account  of  the  present  status  of  the  theory  of  algebraic  continued 
fractions.  At  the  close  of  the  next  lecture  a  bibliography  of 
memoirs  connected  with  the  subject  is  appended,  to  which  refer 
ence  is  made  throughout  this  lecture  and  the  next  by  means  of 
numbers  enclosed  in  square  brackets. 

By  the  term  algebraic  continued  fraction  is  understood,  in  dis 
tinction  from  a  continued  fraction  with  numerical  elements,  one 
in  which  the  elements  —  i.  e.,  the  partial  numerators  and  denomi 
nators  —  are  functions  of  a  single  variable  x  or  of  several  varia 
bles  [16,  a,  p.  4].  Although  the  term  algebraic  does  not  seem  to 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     135 

me  to  be  fortunately  chosen,  I  shall  nevertheless  accept  it  and  use 
it  to  indicate  the  class  of  continued  fractions  which  it  is  proposed 
to  consider  here. 

The  first  foundations  of  a  theory  of  continued  fractions  were 
laid  by  Eider,  who  early  employed  them  [1,  a]  to  derive  from  a 
given  power  series 


a  continued  fraction  of  the  form 

1  a,x  a9x 

(1)  b~+bx+d  +  b~^+J  +'"' 

A  second  form,  also  introduced  by  Euler  *  [46,  a]   is  the  more 
familiar  one 


1  +    1    +    1    +    1   + 


which  was  later  used  by  Gau-ss  [34]  in  his  celebrated  continued1 
fraction  for  F(a,  /3,  7,  x)jF(a,  £  +  1,  7  +  1,  ar).  From  this  time- 
on  still  other  forms  were  discovered  so  that  it  became  impossible 
to  speak  of  a  unique  development  of  a  function  into  a  continued! 
fraction.  Among  these  forms  may  be  especially  mentioned  the 
continued  fraction 


(3') 


a.x  +  b,  -f  ajc  -f  62  -j-  a..x  -f  6  -j- 


used  by  Heine,  Tchebychef,  and  others  in  approximating  to  series 
in  descending  powers  of  x.  By  the  substitution  of  I/a?  for  x  and 
a  simple  reduction  this  can  be  transformed,  after  the  omission  of  a 
factor  xy  into 

i  r2  2 

(3) 

at  +  b^  +  «2  +  b2x  +  ayi?  +  63  +    ' 

The  reason  for  this  variety  of  form  and  for  the  occurrence,  in 

*Pade  in  his  thesis  (p.  38)  traces  it  back  to  Lambert  [2,  «]  and  Lagrange, 
but  Baler's  u?e  is  earlier  still. 


136  THE  BOSTON  COLLOQUIUM. 

particular,  of  the  three  types  just  given  is  discussed  by  Pade  in 
his  thesis  [16,  a] .     As  this  thesis  is  the  foundation  for  a  systematic 
study  of  continued  fractions,  it  will  be  necessary  to  give  a  recapit 
ulation  of  its  chief  results. 
Let 

(4)  %)-v+v +V+---*  (co  =  i) 

be  any  given  power  series,  whether  convergent  or  divergent.  If 
Np(x)jDq(x)  denotes  an  arbitrary  rational  fraction  in  which  the 
numerator  and  denominator  are  of  the  pth  and  gth  degrees  respec 
tively,  there  will  be  p  -f  q  +  1  parameters  which  can  be  made  to 
satisfy  an  equal  number  of  conditions.  Let  them  be  so  determined 
that  the  expansion  of  NJDq  in  ascending  powers  of  x  shall  agree 
with  (4)  for  as  great  a  number  of  terms  as  possible.  In  general, 
we  can  equate  to  zero  the  first  p  -f-  q  +  1  coefficients  of  the  expan 
sion  of  DqS(x)  —  Np  in  ascending  powers  of  x,  and  no  more. 
Hence,  unless  Np  and  Dq  have  a  common  divisor,  the  series  for 
NJDq  agrees  with  (4)  for  an  equal  number  of  terms,  and  the 
approximation  is  said  to  be  of  the  (p  -+-  q  -\-  l)th  order.  In  excep 
tional  cases  the  order  of  the  approximation  may  be  either  greater 
or  less.  Pade  examines  these  exceptional  cases  and  proves  strictly 
that  among  all  the  rational  fractions  in  which  the  degrees  of  numer 
ator  and  denominator  do  not  exceed  p  and  q  respectively,  there 
is,  taken  in  its  lowest  terms,  one  and  only  one,  the  expansion  of 
which  in  a  series  will  agree  with  (4)  for  a  greater  number  of  terms 
than  any  other.  Such  a  rational  fraction  I  shall  term  an  approxi- 
vnard  of  the  given  series. 

The  existence  of  approximants  was,  of  course,  well  known 
before  Pade,  but  no  systematic  examination  of  them  had  been 
\made  except  by  Frobenius  [13],  who  determined  the  important 
relations  which  normally  exist  between  them.  Pade  goes  further, 
and  arranges  the  approximants,  expressed  each  in  its  lowest  terms, 
into  a  table  of  double  entry  : 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    137 


f»=l 


iC  =  c° 

§-cc+ciaJ 

_ 

^2°    =  (*0  +   V  +  C2^ 

s 

AU 

^21 

Dn 

4 

^ 

N02 

^12 

^22 

JL 

D 

^22 

=  3 


When  the  order  of  approximation  of  a  rational  fraction,  taken 
in  its  lowest  terms,  is  exactly  equal  to  the  sum  of  the  degrees  of 
numerator  and  denominator,  increased  by  1,  the  fraction  will  be 
found  once  and  only  once  in  the  table.  If,  conversely,  a  fraction 
^pq/^pq  occurs  but  once  in  the  table,  the  numerator  and  denomi 
nator  are  of  degree  p  and  q  respectively,  and  the  order  of  the 
approximation  which  the  fraction  affords  is  exactly^?-}-  <?-f-l. 
The  approximant  is  then  said  by  PacU  to  be  normal.  AVe  shall 
also  call  the  table  normal  when  it  consists  only  of  normal  fractions, 
or,  in  other  words,  when  no  approximant  occurs  more  than  once  in 
the  table. 

Obviously  all  approximants  which  lie  upon  a  line  perpendicular 
to  the  principal  diagonal  of  the  table  correspond  to  the  same 
value  of  p  -f  q  -f  1.  Hence  in  a  normal  table  they  approximate 
to  (4)  in  equal  degree,  and  accordingly  may  be  said  to  be  equally 
advanced  in  the  table.  If  p  -f  q  +  1  increases  in  passing  from 
one  fraction  to  another,  the  latter  is  the  more  advanced. 

Two  approximauts  will  be  called  contiguous  if  the  squares  of 
the  table  in  which  they  are  contained  have  either  an  edge  or  a 
vertex  in  common. 

Consider  now  a  normal  table,  and  take  any  succession  of  approx 
imants,  beginning  with  one  upon  the  border  of  the  table  and  pass 
ing  always  from  one  approximant  to  another  which  is  contiguous 
to  it  but  more  advanced.  Pad&  shows  that  any  such  sequence  of 
approximants  makes  a  continued  fraction  of  which  the  approxi- 


138  THE  BOSTON  COLLOQUIUM. 

mants  are  the  successive  convergents.  *  Thus  a  countless  manifold 
of  continued  fractions  can  be  formed,  any  one  of  which  through 
its  convergents  gives  the  initial  series  to  any  required  number  of 
terms  and  hence  defines  the  series  and  table  uniquely.  In  all  of 
Pade's  continued  fractions  the  partial  numerators  are  monomials 
in  x. 

The  continued  fraction  is  called  regular  when  its  partial  numer 
ators  are  all  of  the  same  degree  and  likewise  its  denominators, 
certain  specified  irregularities  being  admitted  in  the  first  one  or 
two  partial  fractions.  These  irregularities  disappear  when  the 
continued  fraction,  as  is  most  usual,  commences  with  the  corner 
element  of  the  table.  (Cf.  the  continued  fractions  (2)  and  (3).) 

In  a  normal  table  a  regular  continued  fraction  can  be  obtained 
in  any  one  of  three  ways.  If  we  take  for  the  convergents  the 
approximants  which  fill  a  horizontal  or  vertical  line,  a  continued 
fraction  is  obtained  which  —  except  for  the  irregularity  permitted 
at  the  outset — is  of  the  form  (1)  given  above.  If  the  approxi 
mants  lie  upon  the  principal  diagonal  or  any  parallel  line,  the  con 
tinued  fraction  is  of  type  (3).  Lastly,  if  the  convergents  lie  upon 
a  stair-like  line,  proceeding  alternately  one  term  horizontally  to 
the  right  and  one  term  vertically  downward,  the  continued  fraction 
is  of  the  familiar  form  (2). 

When  a  table  is  not  normal,  the  approximants  which  are  iden 
tical  with  one  another  are  shown  by  Fade  to  fill  always  a  square, 
the  edges  of  which  are  parallel  to  the  borders  of  the  table.  When 
the  square  contains  (n  -f  I)2  elements,  the  irregularity  may  be  said 
to  be  of  the  nth  order.  The  vertical,  horizontal,  diagonal  and 
stair-like  lines  give  regular  continued  fractions  as  before,  unless 
they  cut  into  one  or  more  of  these  square  blocks  of  equal  approxi 
mants.  When  this  happens,  certain  irregularities  appear  in  the 
continued  fraction  which  give  rise  to  various  difficulties  in  the 
consideration  of  matters  of  convergence  and  other  questions. 

On  this  account  it  is  natural  to  inquire  first  whether  the  con 
tinued  fraction  has  or  has  not  a  normal  character.  If  it  has,  the 

*This  is  also  tacitly  implied  in  the  relations  given  by  Frobenius  [13,  p.  5]. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     139 

existence  of  the  three  regular  types  of  continued  fractions  is  as 
sured.  The  necessary  and  sufficient  condition  that  the  table  shall 
be  normal  is  that  no  one  of  the  determinants 


(a,  £^0; 

c.  =  0  if  i  <  0) 


shall  vanish  [16,  a,  p.  35].  It  will  be  noticed  that  the  determi 
nants  are  of  the  same  sort  as  those  which  play  so  conspicuous  a 
role  in  ffadamard's  discussion  of  series  representing  functions 
with  polar  singularities. 

So  far  as  I  am  aware,  the  normal  character  of  the  table  has 
been  established  as  yet  only  in  the  following  cases:  (1)  for  the 
exponential  series  [37]  and  for  (1  -f  x)m  when  m  is  not  an  integer 
[35,  d]  ,  f  by  Fade  ;  and  (2)  for  the  series  of  Stidtje*,  by  myself  [45]  . 

The  construction  ofPad&s  table  leads  at  once  to  a  number  of  new 
and  important  questions.  The  numerators  and  the  denominators  of 
the  approximants  constitute  groups  of  polynomials  which  it  is  only 
natural  to  expect  will  be  characterized  by  common  or  kindred 
properties.  The  table  then  affords  a  suitable  basis  for  the  classifi 
cation  of  polynomials.  Thus,  for  example,  the  polynomials  of 

f  At  least  half  of  the  table  forF(a,  1,  y,  x)  has  a  normal  character.  This  was 
proved  incidentally  in  my  thesis  [76]  by  showing  that  the  remainders  corre 
sponding  to  approximants  on  or  above  the  diagonal  of  the  table  were  all  distinct. 
The  method  of  conformal  representation  was  there  employed,  but  the  same  fact 
can  also  be  demonstrated  very  simply  by  means  of  Gauss'  relationes  inter  contiguas 
(formulas  (19)  and  (20)  of  [34]).  The  approximants  in  the  other  half  of  my 
table  (Cf.  [76],  p.  44)  were  constructed  on  different  principles  from  Fade's, 
the  approximation  being  made  simultaneously  with  reference  to  two  points, 
x  =  0and  x=oo,  but  the  resulting  continued  fractions  were  of  the  same  form 
as  Padd's.  It  is  noteworthy  that  the  relationes  inter  contiguas  lead  to  such  a 
table  rather  than  to  the  one  of  Fade's  construction. 

In  the  case  of  F(—  m,  1,  1,  —  z)  =  (l  +  x)m  the  half  of  Fade's  table  below  the 
diagonal  is  also  normal,  since  the  reciprocal  of  the  approximants  in  the  lower 
half  are  the  approximants  in  the  upper  half  of  the  table  for 


JK  1,1,  -*)  =  (!  +  *)-».    - 
The  normal  character  of  the  table  for  e*  then  follows  since  e^lim  F(gt  1,  1,  z/0). 


140  THE  BOSTON  COLLOQUIUM. 

Legendre  and  similar  polynomials  are  obtained  from  the  series  for 
log  (1  —  #)/(!  4-  x\  while  the  numerators  and  denominators  of  the 
approximants  for  (1  +  a?)"1  are  the  hypergeometric  polynomials 
f\  —  /-t,  —  v  db  m,  —  /A  -\-  v,  —  .T),  in  which  ft  and  z>  are  integers,  or 
the  so-called  polynomials  of  Jacobi  [65] .  In  these,  as  in  numerous 
other  cases,  the  denominators  of  the  convergents  and  the  remainder- 
functions,*  formed  by  multiplying  each  denominator  into  the  cor 
responding  remainder,  are  solutions  of  homogeneous  linear  dif 
ferential  equations  of  the  2nd  order  which  have  a  common  group, 
and  the  relations  of  recurrence  between  three  successive  denomi 
nators  or  remainder-functions  are  the  relationes  inter  contiguas  of 
Gauss  and  Riemann.  (See  in  particular,  [75,  c?]  and  [76].) 
The  further  study  of  such  groups  of  polynomials  will  probably 
bring  to  light  new  and  important  properties.  The  position  of 
the  roots  of  the  denominators  should  especially  be  ascertained,  be 
cause  the  distribution  of  these  roots  has  an  intimate  connection  with 
the  form  of  the  region  of  convergence  of  the  continued  fraction 
and  oftentimes  also  with  the  position  and  character  of  the  function 
which  the  continued  fraction  defines. 

Probably  the  most  fundamental  question  concerning  Pad&s 
table  is  that  of  the  convergence  of  the  various  classes  of  continued 
fractions  or  lines  of  approximants.  The  first  investigation  of  the 
convergence  of  an  algebraic  continued  fraction  was  made  by  Rie 
mann  [18]  in  1863,  followed  by  TJiome  [19]  a  few  years  later,  f 
Both  writers  investigated  the  continued  fraction  of  Gauss  by 
rather  painful  methods,  not  based  absolutely  upon  the  algo 
rithm  of  the  continued  fraction  but  upon  extraneous  considera 
tions.  This  is  not  surprising,  for  there  were  at  that  time  no  gen 
eral  criteria  for  the  convergence  of  continued  fractions  with 
complex  elements,  and  even  now  the  number  is  astonishingly 
small. 


*  In  at  least  half  of  the  table.     See  the  preceding  footnote. 

t  As  Riemann's  work  appeared  posthumously,  Thome's  has  the  priority  of 
publication  (1866)  but  was  itself  preceded  by  Worpitzky's  dissertation,  to  which 
reference  is  made  in  a  subsequent  footnote. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     141 

The  two  principal  criteria  for  convergence   correspond  to  the 
familiar  tests  for  the  convergence  of  a  real  continued  fraction 


2 


in  which  either  (1)  all  the  elements  are  positive  or  (2)  the  partial 
denominators  X.  are  positive  and  the  partial  numerators  fi.  are 
negative.  The  latter  class  of  real  continued  fractions  is  known  to 
converge  if  Xt.=  l  —  p..  Pringsheim  [29]  has  shown  that  when 
the  elements  are  complex,  the  condition  |  V  =  1  -+•  p.  is  still 
sufficient  for  convergence.  If,  furthermore,  the  continued  frac 
tion  has  the  customary  normal  form  in  which  pn  =  1  ,  the  condition 
may  be  replaced  by  the  less  restrictive  one  [29,  p.  320], 

1 


\» 


The  necessary  and  sufficient  condition  for  the  convergence  of  the 
first  class  of  real  continued  fractions  can  be  most  easily  expressed 
after  it  has  been  reduced  to  the  form 


If  then  2X^  is  divergent,  the  continued  fraction  converges,  while 
it  diverges  if  2X^  is  convergent.*  But  in  the  latter  case  limits  exist 
for  the  even  and  the  odd  couvergeuts  when  considered  separately. 
This  result  is  included  in  the  following  theorem  which  I  gave  in 
the  Transactions  of  1901  for  continued  fractions  with  complex 
elements  [31]  :  If  in 

1  1  1 

^  +  i^  +  a2  +  i/32  +  <vMiS3  +  ' 

the  elements  a.  have  all  the  same  sign  and  the  /3.  are  alternately 
positive  and  negative,  f  the  continued  fraction  will  converge  if 
2  1  an  +  ipn  |  is  divergent  ;  on  the  other  hand,  if  2  an  +  ij3n  \  is 


*Seidel,  Habilitationsschrift,  1846,  and  Stern,  Journ.fdr  Math.,  vol.  37  (1848), 
p.  269. 

fZero  values  are  permissible  for  either  at  or  3,. 


142  THE  BOSTON  COLLOQUIUM. 

convergent  and  either  the  a.  or  the  /3.  fulfill  the  condition  just 
stated  concerning  their  signs,  the  even  and  the  odd  convergents 
have  separate  limits. 

The  most  general  criterion  for  the  convergence  of 

*!  *2          &S 

1    +    1    +   1     +    ' 

(6i  real  or  complex)  seems  to  be  the  one  which  I  gave  in  October, 
1901  [32,  6,  §5]. 

Two  remarks  of  a  general  nature  concerning  the  convergence 
of  algebraic  continued  fractions  may  be  of  interest.  In  the  con 
sideration  of  numerical  continued  fractions  a  difficulty  frequently 
encountered  is  that  the  removal  of  a  finite  number  of  partial 
fractions  ^{l\  at  the  beginning  of  (5)  may  affect  its  convergence 
or  divergence.  The  convergence  is  therefore  not  determined 
solely  by  the  ultimate  character  of  the  continued  fraction,  as  is 
true  of  a  series.  Pringsheim[29~\  has  proposed  to  call  the  con 
vergence  unconditional  when  it  is  not  destroyed  by  the  removal 
of  the  first  n  partial  fractions  of  (5).  The  difficulties  due  to  con 
ditional  convergence  usually  disappear  from  consideration  in  treat 
ing  algebraic  continued  fractions.  For  let  NnjDn  now  denote 
the  nth  convergent.  If  after  the  removal  of  the  first  n  partial 
fractions  the  continued  fraction  converges  uniformly  in  a  given 
region  and  accordingly  represents  a  function  F(z)  which  is  holo- 
morphic  within  the  region,  then  after  the  restoration  of  the  initial 
terms  the  continued  fraction  will  define  the  function 


™ 


which  must  be  either  holomorphic  or  meromorphic  within  the  given 
region  [32,  a  or  c]  .  An  exception  occurs  only  if  the  denominator 
of  (6)  vanishes  identically  in  the  region.  This  is  impossible  for 
the  second  and  third  types  of  continued  fractions,  since  the  de 
velopment  of  a  rational  fraction  —  DJDn_l  in  either  type  (2) 
or  (3)  consists  of  a  finite  number  of  terms,  whereas  the  develop 
ment  of  F(z\  by  hypothesis,  continues  indefinitely. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     143 

The  second  remark  relating  to  convergence  is  that  its  discus 
sion  for  a  continued  fraction  is  usually  reduced  to  the  correspond 
ing  question  for  an  infinite  series.  The  succession  of  convergents 


is,  in  fact,  obviously  equivalent  to  the  series 

\ 
-< 


But  the  latter  by  means  of  the  familiar  relations  connecting  the 
denominators  or  the  numerators  of  three  consecutive  convergents 
may  be  reduced  to  the  form  : 


Dn+lDn+2 


+1» 


We  turn  now  from  these  general  considerations  to  the  questions 
of  convergence  connected  with  Pade's  table.  Under  what  con 
ditions  will  the  various  lines  of  approximants  converge ;  in  par 
ticular,  the  three  standard  types  of  continued  fractions  obtained 
by  following  (1)  the  horizontal  or  vertical  lines,  (2)  the  stair-like 
lines,  and  (3)  the  diagonal  lines  ?  When  they  converge  simul 
taneously,  have  they  a  common  limit?  If  not,  what  are  the 
mutual  relations  between  the  functions  which  they  define?  What 
is  the  form  of  the  region  of  convergence? 

These  and  other  questions  press  upon  us,  and  are  of  great  in 
terest.  A  complete  investigation  has  been  made  only  for  the 
exponential  series.  Fade  [37,  a]  finds  that  when  p/q  for  any  suc 
cession  of  approximants  N^D^  converges  to  a  value  o>,  the  ap 
proximants  converge  toward  the  generating  function  e*  for  all 
values  of  .r.  Furthermore,  the  numerators  and  denominators  sepa 
rately  converge,  the  former  to  the  limit  e"x/t»~l,  the  latter  to  e~x/<a+l. 
This  smooth  result  is  not,  however,  a  typical  one,  not  even  for 
entire  functions.  It  is  due  at  least  in  part  to  the  fact  that  e?  is 


144  THE  BOSTON  COLLOQUIUM. 

an  entire  function  without  zeros.  This  will  be  apparent  after  an 
examination  has  been  made  of  the  vertical  and  horizontal  lines 
of  Fade's  table,  which  we  now  proceed  to  consider. 

It  is  obvious  that  the  first  p  -\-  q  +  \  terms  of  the  given  series 
(4)  determine  an  equal  number  of  terms  of  the  series  for  its  re 
ciprocal.  If,  therefore,  in  the  table  each  approximant  is  replaced 
by  its  reciprocal  and  the  rows  and  columns  are  then  interchanged, 
we  shall  obtain  the  table  for  the  reciprocal  series.  The  problems 
presented  by  the  horizontal  and  vertical  lines  of  the  table  are  con 
sequently  of  essentially  the  same  character,  and  our  attention  may 
be  confined  henceforth  to  the  horizontal  lines  alone. 

By  the  interchange  just  described  the  zeros  and  poles  of  (4) 
become  the  poles  and  zeros  respectively  of  the  reciprocal  function. 
In  the  case  of  the  exponential  function  the  reciprocal  series  has 
the  same  character  as  the  initial  series,  each  defining  an  entire 
function  without  zeros,  and  the  simultaneous  convergence  of  rows 
and  columns  for  all  values  of  x  was  therefore  to  be  expected  ;  but 
in  general  this  does  not  hold. 

In  investigating  the  convergence  of  the  horizontal  lines  the  first 
case  to  be  considered  is  naturally  that  of  a  function  having  a  number 
of  poles  and  no  other  singularities  within  a  prescribed  distance  of 
the  origin.  It  is  just  this  case  that  Montessus  [33,  a]  has  exam 
ined  very  recently.  Some  of  you  may  recall  that  four  years  ago  in 
the  Cambridge  colloquium  Professor  Osgood  *  took  Hadamard's 
thesis  f  as  the  basis  of  one  of  his  lectures.  This  notable  thesis  is 
devoted  chiefly  to  series  defining  functions  with  polar  singularities. 
Montessus  builds  upon  this  thesis  and  applies  it  to  a  table  possess 
ing  a  normal  character.  Although  his  proof  is  subject  to  this 
limitation,  his  conclusion  is  nevertheless  valid  when  the  table  is 
not  normal,  as  I  shall  show  in  some  subsequent  paper. 

The  first  horizontal  row  of  the  table  scarcely  needs  considera 
tion,  for  it  consists  of  the  polynomials  obtained  by  taking  suc 
cessively  1,  2,  3,  •  •  •  terms  of  the  series.  Consequently  the  con 
tinued  fraction  obtained  from  the  first  row, 

*BulL  of  the  Amer.  Math.  Soc.,  vol.  5,  pp.  74-78. 
f  Journ.  de  Math.,  ser.  4,  vol.  8  (1892). 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    145 


1   —  a^  +  «0  —  a2x  -f  a.j  —  a3#  +  a2  — 

is  identical  with  the  series,  and   its  region  of  convergence  is  a 
circle. 

Let  El  be  the  radius  of  this  circle  and  ql  the  number  of  poles 
of  (4)  which  lie  upon  its  circumference.  Suppose  also  that  the  next 
group  of  poles,  q2  in  number,  lie  upon  a  circle  of  radius  R2,  hav 
ing  its  center  in  the  origin  ;  that  <?3  poles  lie  upon  the  next  circle 
(R3) ;  and  so  on  indefinitely  or  until  a  circle  is  reached  which  con 
tains  a  non-polar  singularity.  Hadamard  (L  c.,  §  18)  has  proved 
that  the  denominators  Dpq  of  the  approximants  of  the  (ql  +  l)th 
row,  of  the  (ql  -f-  q2  -f-  l)th  row,  and  so  on,  approach  a  limiting 
form  as  we  advance  in  the  row,  and  that  the  limiting  polynomials 
give  the  positions  of  the  first  qv  ql  +  q2)  •  •  •  poles  respectively.. 
Thus  if,  for  example, 


and 


Km  £<<>  = 


I 

the  first  group  of  poles  are  the  roots  of  the  polynomial 
1  -f  B}x  +  •  •  •  Bqixqi.  Using  this  result  of  Hadamard,  Montes- 
sus  shows  that  in  a  normal  table  the  approximants  of  the  ((^-f  l)th 
tow  converge  at  every  point  within  the  circle  (E2)  —  excepting, 
of  course,  at  the  ql  poles  —  but  not  without  this  circle  ;  that  the 
approximants  of  the  (^  -f-  q2  -f  l)th  row  converge  similarly  within 
the  circle  (JF?3)  except  at  the  included  ql  -f  q2  poles ;  and  so  on. 
In  proving  this  Montessus  makes  use  of  an  idea  advanced  in 
Fade's  thesis  ([16,  a,  p.  51] ,  or  [24])  which,  though  applicable  in 
the  present  case,  is  possibly  somewhat  misleading.  In  Fade's  con 
tinued  fractions  the  partial  numerators  /LL  are  monomials  in  x.  This 
is  due  to  the  fact  that  there  is  a  steady  increase  in  the  order  of  the 
approximation  afforded  by  the  successive  convergents  at  x  =  0. 
Consider  now  the  series  (7),  and  let  T  denote  the  region  or  set  of 
points  in  the  avplane  for  which  |  Dn  ,  from  and  after  some  value 
of  n,  has  both  an  upper  and  a  lower  limit  Then  in  T  the  con- 

i 


146  THE  BOSTON  COLLOQUIUM. 

tinued  fraction  will  converge  or  diverge  simultaneously  with  the 
power  series, 

(8)  /*„+!   -  /*„+ A+2   +  A*n+i/*B+2^«+3 

Call  C  the  circle  of  convergence  of  (8).  At  all  points  of  T 
within  O  the  continued  fraction  converges,  and  at  all  exterior 
points  of  T  it  diverges.  On  this  account  Fade  proposes  to  call 
C  the  "  circle  of  convergence  "  of  the  continued  fraction.  In  the 
case  which  we  have  just  been  discussing  this  concept  is  applicable 
because  of  the  existence  of  limiting  forms  for  the  denominators  of 
the  rows  considered.  The  region  T  comprises  the  entire  finite 
plane  with  the  exception  of  the  roots  of  the  limiting  form,  and 
the  circle  C  is  successively  identical  with  (R2),  (-&3)>  ....  Thus, 
as  we  pass  down  the  rows  of  the  table,  we  obtain  continued  frac 
tions  having  an  increasing  region  of  convergence. 

In  introducing  the  term  circle  of  convergence  for  a  continued 
fraction  Fade  ignores  all  points  not  included  in  T.  Call  the  ex 
cluded  point-set  T'.  If  Dn  \  increases  indefinitely  with  increas 
ing  n  over  the  whole  or  a  part  of  T  the  series  (7)  may  converge, 
and  this  may  happen  even  though  (8)  is  a  divergent  series.*  The 
term  circle  of  convergence  is  therefore  an  inappropriate  one,  al 
though  the  considerations  upon  which  it  is  based  are  useful. 

Nothing  more  of  account  seems  to  be  known  concerning  the 
the  convergence  of  the  horizontal  and  vertical  lines. f  The  more 
common  and  important  continued  fractions  are  obtained  from 
diagonal  and  stair-like  path&  through  the  table.  In  many  familiar 
continued  fractions  of  the  second  type, 

a0      a,x      a2x      a^x 

T+  V+T  +  T  +  '"' 


*The  coefficients  in  the  continued  fraction  of  Stieltjes  (discussed  later  in  the 
lecture)  can  be  easily  so  determined  as  to  give  a  case  of  this  sort,  the  region  of 
convergence  of  (7)  being  the  entire  plane  with  the  exception  of  the  negative 
half  of  the  real  axis.  We  suppose,  with  Fade  that  the  absolute  term  of  Dn  is 
taken  equal  to  1. 

t  It  is  perhaps  worth  noting  that  the  coefficients  in  the  first  type  of  continued 
fractions  can  not  be  selected  arbitrarily  if  it  is  to  be  connected  with  such  a  table 
as  Fade  constructs.  In  the  other  two  types  the  coefficients  are  entirely  arbitrary. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     147 

an  with  increasing  n  approaches  a  limit,  as  for  instance  in  the  con 
tinued  fraction  of  Gams  where  lim  an  =  —  J.  The  significance  of 
the  existence  of  such  a  limit  I  first  pointed  out  for  a  comprehen 
sive  class  of  cases  in  1901  [32,  «],  and  since  then  I  have  shown 
by  simpler  methods  [32,  c]  that  the  result  is  perfectly  general. 
Let  lim  an  =  k.  Then  the  continued  fraction  converges,  save  at 
isolated  points,  over  the  entire  plane  of  x  with  the  exception  of 
the  whole  or  a  part  of  a  cut  drawn  from  x  =  —  1  /4k  to  x  =  oo  in 
a  direction  which  is  a  continuation  of  the  vector  from  x  =  0  to 
x  =  —  1  /4A-.  Within  the  plane  thus  cut  the  limit  of  the  continued 
fraction  is  holomorphic  except  at  the  isolated  points  which  (if  they 
exist)  are  poles.  When  there  is  no  limit  for  an  but  only  an  upper 
limit  U  for  its  modulus,  the  continued  fraction  (see  [32,  6])  is  mero- 
morphic  or  holomorphic  at  least  within  a  circle  of  radius  1/4U 
having  its  center  in  the  origin.*  A  special  case  is  that  in  which 
lim  an  =  0.  The  limit  of  the  continued  fraction  is  then  a  function 
which  is  holomorphic  or  meromorphic  over  the  entire  plane.  A 
comparison  of  this  last  result  with  that  of  MonUssus  shows  that  a 
much  greater  region  of  convergence  has  now  been  obtained.  This 
is  doubtless,  in  general,  a  reason  for  preferring  the  second  and 
third  types  of  continued  fractions  to  the  first. 

As  another  illustration  of  the  second  type  of  continued  fraction 
I  shall  choose  the  celebrated  continued  fraction  of  Stieltjes  [26,  a]. 
In  this  each  coefficient  an  is  positive.  By  putting  x  =  1/2  in 
(2),  the  continued  fraction,  after  dropping  a  factor  z,  can  be  thrown 
into  the  form 

a[z  -f-  a/2  -f  a^z  -f  a[-f  a'5z+  '  " '  ^/fl>    '* 

which  is  the  form  preferred  by  Stieltjes.     To  every  such  con 
tinued  fraction  there  corresponds  a  series 

*  A  demonstration  of  this  property  within  the  circle  (1/4£T)  has  been  pre 
viously  given  in  a  dissertation  by  Worpitzky  [18  bis],  which  has  come  to  my 
notice  for  the  first  time  during  the  examination  of  the  proof-sheets  of  these  lec 
tures.  This  dissertation  bears  the  date  1865  and  appears  to  be  the  earliest  pub 
lished  memoir  treating  of  the  convergence  of  algebraic  continued  fractions. 


148 


THE  BOSTON  COLLOQUIUM. 


(9) 

for  which 


A  = 


(10) 


C0         Cl 

C2           ••'     Cn-l 

Cl          C2 

C3           "  '    Cn          \ 

I 

.  .  .  .      .  i 

$ 

Cn-l    C, 

,        C»+l      •••C2n-2! 

The  correspondence  is  also  a  reciprocal  one.  To  every  series 
which  fulfills  these  conditions  there  corresponds  a  continued  frac 
tion  of  the  above  type  with  positive  coefficients.  From  the  con 
ditions  (10)  it  follows  that  c^>0  and  that  Gn/cn_l  >  cn_Jcn_r 
If,  therefore,  the  increasing  ratio  cn/cn_1  has  a  finite  limit,  the 
series  is  convergent.  On  the  other  hand,  if  it  increases  without 
limit,  the  series  is  divergent. 

In  investigating  the  convergence  of  the  continued  fraction  the 
especial  skill  of  Stieltjes  was  shown.  From  the  relation  connect 
ing  three  consecutive  denominators  (numerators)  of  the  conver- 
gents  it  was  shown  easily  that  either  set  of  alternate  denominators 
(numerators)  made  a  Sturm's  series,  whence  it  follows  that  all  the 
roots  of  the  denominators  (numerators)  lie  upon  the  negative  half 
of  the  real  axis  of  z.  This  leads  naturally  to  the  conjecture  that 
the  region  of  convergence  will  be  the  entire  plane  of  z  with  the 
exception  of  the  whole  or  a  part  of  the  negative  half  axis,  and 
that  the  functional  limit  will  have  no  zeros  exterior  to  this  half 
of  the  axis.  First  the  convergence  is  examined  when  z  is  real 
and  positive.  The  criterion  of  SMd,  cited  previously  in  this 
lecture,  then  applies.  If,  namely,  2an  is  divergent,  the  continued 
fraction  will  converge  along  the  positive  axis,  while  if  2aB  is  con- 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     149 

vergent,  the  two  sets  of  alternate  convergents  have  limits  which 
are  distinct.  The  conclusion  is  next  extended  by  Stieltjes  to  the 
half  of  the  complex  plane  for  which  the  real  part  of  z  is  positive. 

This  brings  him  to  the  difficult  part  of  his  problem,  the  exten 
sion  of  the  result  to  the  other  half-plane  but  with  exclusion  of  the 
real  axis.  Here,  particularly,  Stieltjes  [26,  a,  §  30]  shows  his 
ingenuity.  He  overcomes  the  difficulty  by  establishing  first  a 
preliminary  theorem  which  is  of  vital  importance  for  sequences 
of  polynomials  or  rational  fractions.  The  theorem  is  as  follows. 
Let  /!(»),  /2(z),  •  •  •  be  a  sequence  of  functions  which  are  holomor- 
phic  within  a  given  region  T,  and  suppose  that  2*=]/n(z)  is  uni 
formly  convergent  in  some  part  T"  of  the  interior  of  T.  Then 
if  f^z)  +fjz)  -f  •  •  •  +fn(z)  has  an  upper  limit  independent  of  n  in 
any  arbitrary  region  T'  which  includes  T"  but  is  contained  in  the 
interior  of  Ty  the  series  2/n(z)  will  converge  uniformly  in  T'  and 
therefore  has  as  its  limit  a  function  which  is  holomorphic  over  the 
whole  interior  of  T*. 

In  the  application  of  this  theorem  Stieltjes  decomposes  each 
convergent  Nn(z)/Dn(z)  into  partial  fractions, 

M.  Mz  M 

-  L_   -1-    -  ?__L    ----  L  -  r— 

z  -f  a-j      z  +  a2  z  -f  ar 

in  which 

J/(>0,         a,  SO,          iX  =  c, 

i=l 

From  this  it  follows  that  NJDn  has  an  upper  limit  independent 
of  n  in  any  closed  region  of  the  plane  which  does  not  contain  a 
point  of  the  negative  half-axis.  If  now  in  either  the  sequence 
of  the  odd  convergents  or  of  the  even  convergents  we  denote  the 
nth  term  of  the  sequence  by  NnjDn  and  place 


the  series  2*=1/n(a)  converges  uniformly  in  any  portion  of  the  plane 

*For  a  further  extension  of  this  line  of  work,  see  Osgood,  Annals  of 
ser.  2,  vol.  3(1901),  p.  25. 


150  THE  BOSTON  COLLOQUIUM. 

for  which  the  real  part  of  »  is  positive.  All  the  conditions  of  the 
lemma  of  Stieltjes  are  now  fulfilled,  and  the  region  of  convergence 
may  be  extended  over  the  entire  plane  with  the  exception  of  the 
negative  half-axis. 

On  account  of  the  uniform  character  of  the  convergence  the 
limit  of  either  sequence  is  holomorphic  at  every  point  exterior  to 
the  negative  half-axis.  When  2a^  is  divergent,  the  two  limits 
coincide  and  the  continued  fraction  itself  is  convergent.  On  the 
other  hand,  if  2c^  is  convergent,  the  two  limits  are  distinct. 
Stieltjes  shows  also  that  in  the  latter  case  the  numerators  and  the 
denominators  of  either  sequence  converge  to  holomorphic  functions 
P(Z)>  <l(z)  °f  9enre  0,  and  the  two  pairs  of  functions  are  connected 
by  the  equation 


which  corresponds  to  the  familiar  relation 


A  more  direct  method  [31]  of  demonstrating  the  convergence 
results  of  Stieltjes  is  by  an  extension  *  of  the  criterion  previously 
cited  for  the  convergence  of  continued  fractions  in  which  the 
partial  fractions  !/(«„  4-  *'$„)  have  an  an  of  constant  sign  and  a 
@n  of  alternating  sign.  The  introduction  of  the  lemma  of  Stieltjes 
is  consequently  unnecessary,  but  I  wish  nevertheless  to  emphasize 
its  fundamental  importance.  Other  notable  results  which  it  will 
be  impossible  to  reproduce  here  are  also  contained  in  his  splendid 
memoir. 

*  If  namely,  ^  |  an  -f-  i(3n  \  is  divergent  and  the  condition  concerning  the  signs 
either  of  the  an  or  of  the  /?»  is  fulfilled,  the  continued  fraction  will  converge  pro 
vided  |  a*  |/|/3»  |  has  a  lower  or  an  upper  limit  respectively.  Put  now  z—w2  in 
(8X)  so  that  it  becomes 


w  \  a{w  -f-  «2W  ~h 

When  ^a^  is  divergent,  this  falls  under  the  extended  criterion  if  we  put 
o/w  =  an  +  ifin,  except  when  z  is  negative.  On  the  other  hand,  when  ^a'n  is  con 
vergent,  the  criterion  applies  without  extension  directly  to  (8').  In  either  case 
the  uniform  character  of  the  convergence  follows  with  the  addition  of  a  few  lines.  ( 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     151 

It  is  interesting  to  bring  this  work  of  Stieltjes  into  connection 
with  the  table  of  Fade  [44]  .  The  odd  convergents  of  the  con 
tinued  fraction  of  Stieltjes  fill  the  principal  diagonal  of  Pade's 
table,  thus  constituting  by  themselves  a  continued  fraction  of  the 
third  type,  and  the  even  convergents  fill  the  parallel  file  immedi 
ately  below,  forming  a  similar  continued  fraction.  The  signifi 
cance  of  distinct  limits  for  the  two  sets  of  convergents  is  thus 
made  clearer. 

The  series  of  Stieltjes  has  perhaps  its  greatest  interest  when 
treated  in  connection  with  the  theory  of  divergent  series.  Although 
the  continued  fraction  always  converges  if  the  series  does,  the  con 
verse  is  not  true.  For  when  the  series  (9)  is  divergent,  two  cases 
are  to  be  distinguished  according  as  2  a/  is  divergent  or  conver 
gent.  In  the  former  case  the  continued  fraction  gives  one  and 
only  one  functional  equivalent  of  the  divergent  series.  Le  Roy 
states,*  though  without  proof,  that  the  function  furnished  is 
identical  with  the  one  obtained  from  the  series  by  the  method  of 
Borel,  whenever  the  latter  method  is  applicable  also.  When  2c^  is 
convergent,  two  different  functions  are  obtained  from  the  con 
tinued  fraction,  the  one  through  the  even  and  the  other  through 
the  odd  convergents.  And  if  there  are  two  such  functions  which 
correspond  to  the  series,  there  must  be  an  infinite  number.  For 
if  0(.r)  and  ^r(.r),  when  expanded  formally,  give  rise  to  the  same 
divergent  series,  so  also  will 


in  which  c  denotes  an  arbitrary  constant.  Special  properties, 
however,  attach  themselves  to  the  two  functions  picked  out  by  the 
continued  fraction  of  Stieltjes,  upon  which  we  can  not  linger  here. 
This  result  of  Stieltjes  seems  to  me  to  be  especially  significant, 
since  it  indicates  a  division  of  divergent  series  into  at  least  two 
classes,  the  one  class  containing  the  series  for  which  there  is  prop 
erly  a  single  functional  equivalent  and  the  other  comprising  the 

*Loc.  at.,  p.  428. 


152  THE  BOSTON  COLLOQUIUM. 

series  which  correspond  to  sets  of  functions.  It  is,  of  course, 
just  possible  that  this  distinction  may  be  due  to  the  nature  of  the 
algorithm  employed  in  deriving  the  functional  equivalent  of  the 
series,  but  it  is  far  more  probable  that  the  difference  is  intrinsic 
and  independent  of  the  particular  algorithm.  If  this  view  be  cor 
rect,  the  method  of  Borel  which  gives  a  single  functional  equivalent, 
is  limited  in  its  application  to  series  of  the  first  class. 

An  extension  of  the  work  of  Stieltjes  has  been  sought  in  two  dis 
tinct  directions  by  modification  of  the  conditions  imposed  upon  his 
series.  Borel  [43]  so  modifies  them  as  to  make  the  series  (when 
divergent)  fulfill  the  requirement  imposed  in  lecture  2  and  permit  of 
manipulation  precisely  as  a  convergent  series.  In  the  last  number  of 
the  Transactions  *  [45]  I  began  a  study  of  series  which  are  subject  to 
only  one  of  the  two  restrictions  expressed  in  the  inequalities  (10), 
but  was  obliged  to  bring  the  work  to  a  hurried  close  to  prepare  these 
lectures.  In  the  main,  the  corresponding  continued  fractions  have 
the  same  properties  as  the  continued  fraction  of  StieltjeSj  but  a  con 
siderable  difference  is  shown  in  regard  to  convergence.  Though 
the  roots  of  the  numerators  and  denominators  of  the  convergents 
are  still  real,  they  are  no  longer  confined  to  the  negative  half 
of  the  real  axis,  and  may  be  infinitely  thick  along  the  entire 
extent  of  the  axis.  In  certain  cases  the  continued  fraction  con 
verges  in  the  interior  of  the  positive  and  negative  half  planes, 
defining  in  each  an  analytic  function  which  has  the  real  axis  as  a 
natural  boundary.  The  continued  fraction  therefore  effects  the 
continuation  of  an  analytic  function  across  such  a  boundary,  and 
gives  a  natural  instance  of  such  a  continuation  f  —  natural  in  dis 
tinction  from  artificial  examples  set  up  with  the  express  object  of 
showing  the  possibility  of  a  unique,  non-analytic  extension. 

Pade  [17,  a]  has  suggested  the  foundation  of  a  theory  of  diver- 

*July,  1903. 

f  Earlier  instances  of  a  natural  continuation  are  also  to  be  found,  as,  for 
example,  that  afforded  by 


m    m>  (m  -f- 
across  the  axis  of  reals. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     153 

gent  series  upon  the  continued  fractions  of  his  table.  The  diffi 
culties  of  carrying  out  the  suggestion  are  undoubtedly  very  great 
and  have  been  pointed  out  by  BoreL*  Not  only  must  the  con 
vergence  of  the  principal  lines  of  approximants  and  the  agreement 
of  their  limits  be  investigated,  but  the  combination  of  two  or  more 
divergent  series  must  also  be  considered.  It  is  not  enough  to  point 
out,  as  does  Pade,  that  the  approximants  of  given  order  for  any 
two  series,  whether  divergent  or  convergent,  determine  uniquely 
the  approximants  of  the  same  or  lower  order  for  the  sum-  and 
product-series.  For  practical  application  of  the  theory  it  must  be 
proved  also  that  the  function  defined  by  the  table  corresponding 
to  the  newr  series  is,  under  suitable  limitations,  the  sum  or  product 
of  the  functions  defined  by  the  given  divergent  series.  But  great 
as  are  the  difficulties  of  such  an  investigation,  even  for  restricted 
classes  of  series,  the  reward  will  probably  be  correspondingly 
great. 

So  far  as  it  has  been  yet  investigated,  the  diagonal  type  of  con 
tinued  fractions  seems  to  have  accomplished  nearly  everthing  that 
can  fairly  be  asked  of  a  sequence  of  rational  fractions.  Not  only 
does  it  afford  a  convenient  and  natural  algorithm  for  computing 
the  successive  fractions,  but  in  every  known  instance  the  region  of 
convergence  is  practically  the  maximum  for  a  series  of  one  valued 
functions.  The  continued  fraction  offfalphen  [2 1 ,  a] ,  so  frequently 
cited  as  an  instance  of  a  continued  fraction  which  diverges  though 
the  corresponding  series  converges,  might  appear  at  first  sight  to 
be  an  exception.  But  this  divergence  occurs  only  at  special  points. 
In  fact,  the  continued  fraction  not  only  converges  at  the  center  of  the 
circle  of  convergence  for  the  series,  but,  as  Halphen  himself  says, 
continues  the  function  over  the  entire  plane  with  the  exception  of 
certain  portions  of  a  line  or  curve.  If  then,  continued  fractions 
offer  such  advantages  for  known  series  and  classes  of  functions,  is 
it  too  much  to  expect  that  in  the  future  they  will  throw  a  powerful 
searchlight  upon  the  continuation  of  analytic  functions  and  the 
theory  of  divergent  series  ? 

*  Les  Series  divergentes,  p.  60. 


154  THE  BOSTON  COLLOQUIUM. 

LECTURE  6.  The  Generalization  of  the  Continued  Fraction. 
In  the  last  lecture  the  algebraic  continued  fraction  was  presented 
under  the  form  of  a  series  of  approximants  for  a  given  function. 
An  immediate  generalization  of  this  conception  can  be  obtained 
either  by  increasing  the  number  of  points  at  which  an  approxima 
tion  is  sought  or  by  requiring  a  simultaneous  approximation  to 
several  functions.  The  latter  generalization  results  at  once  from 
an  attempt  to  increase  the  dimensions  of  the  algorithm  or,  in  other 
words,  the  number  of  terms  in  the  linear  relation  of  recurrence 
between  the  successive  convergents  or  approximants.  As  this 
generalization  is  without  doubt  the  more  important,  I  shall  make 
it  the  chief  subject  of  this  lecture.  But  a  few  words,  at  least, 
should  be  devoted  to  the  former  extension,  which  is  worthy  of  a 
•  -more  careful  and  systematic  study  than  it  has  received. 

Denote  again  by  Np(x)jD,(x)  a  rational  fraction  with  arbitrary 
coefficients.  These  can,  in  general,  be  so  determined  that  its  ex 
pansion  at  x  —  0  shall  agree  for  nv  successive  terms  with  a  given 
series 

C0  +  CrT  +  C2X2  +    •  •  • 

its  expansion  at  x  —  al  for  n2  successive  terms  with 
60  +  b,(x  -  a,)  +  b.2(x  -  atf  +  ..-, 

at  x  =  a2  for  n3  successive  terms  with 

k0  +  kfo  —  a2)  +  kjx  -  a2)2  +  •  •  • , 

and  so  on,  the  total  number  of  conditions  thus  imposed  being  equal 
to  p  -f  q  -f  1  or  the  number  of  parameters  in  the  rational  frac 
tion.  To  each  set  of  values  for  the  n.  and  q  there  corresponds  an 
approximant,  and  the  various  approximants  can  be  arranged  into 
a  table  of  multiple  entry  according  to  the  values  of  these  quan 
tities.  Continued  fractions,  at  least  in  the  case  of  a  normal  table, 
can  then  be  obtained  by  following  any  path  which  passes  succes 
sively  from  one  approximant  to  another  contiguous  to  it  but  more 
advanced  in  the  table.  As  we  proceed  along  the  path,  the  degree  of 
approximation  for  each  of  the  points  0,  ap  «2,  •  •  •  must  not  decrease 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     155 

while  at  each  step  it  is  to  increase  for  at  least  some  one  point.  The 
partial  numerators  of  the  continued  fraction  are  then  either  posi 
tive  integral  powers  of  x,  x  —  av  x  —  a2,  -  •  •,  or  the  products  of  such 
powers.  The  degrees  of  the  approximations  obtained  by  stopping 
the  continued  fraction  with  any  term  can  be  inferred  readily  from 
the  degrees  of  the  partial  numerators  in  xy  x  —  avx  —  a.2)  •  •  •  .  The 
details  of  the  theory  have  not  been  worked  out.* 

The  interest  of  such  work  can  perhaps  best  be  made  apparent 
by  referring  to  the  developments  for  the  simplest  case  in  which 
each  n.  is  taken  equal  to  1.  The  rational  fraction  NpjDq  is  then 
completely  determined  by  the  requirement  that  at  p  -f  q  4-  1  given 
points  al  =  0,  a2,  ay  -  -  •  it  shall  take  an  equal  number  of  pre 
scribed  values,  Av  A2,  AB,  •  •  •  .  If  these  are  the  values  which  a 
single  function  assumes  at  the  points,  we  have  the  rational  frac 
tions  which  were  introduced  by  Cauchy  into  the  theory  of  inter 
polation  [99,  a]  and  which  have  been  quite  recently  formed  into 
a  table  and  examined  by  Fade  [112].  As  p  +  q  -f  1  increases, 
the  number  of  points  at  which  the  approximation  is  sought  like 
wise  steadily  increases. 

When  q  —  0,  the  rational  fraction  becomes  the  familiar  inter 
polation-polynomial  of  Lagrange, 


in  which 


This  has  been  put  into  a  very  interesting  form  by  Frobenius  [95] 
which  permits,  without  reconstruction,  f  of  an  indefinite  increase  in 
the  number  of  its  terms.  Let  us  first  take  1  j(z  —  x)  as  the  par 
ticular  function  of  x  for  which  an  approximation  is  sought.  From 
the  equations 

*The  only  investigation  of  this  character  is  found  in  [76],  but  on  account  of  the 
nature  of  the  functions  there  considered  certain  variations  were  made  in  the  con 
struction  of  the  table. 

tCf.  also  [99,  a]. 


156  THE   BOSTON   COLLOQUIUM. 

I  1  1  x-a, 


z  —  x      (z  —  aj  —  (x  —  aj       z  —  a^      z  —  a^   z  —  x 

_       *         {x-ai(      *        .,..  a?~ga>       1      \  = 
z  —  «!      z  —  al\z  —  a2      z  —  a2'  z  —  x  ) 


the  series 


/I)    __  |  i          [  2       , 

V'    z  —  x    z—cii      (z  —  al)(z  —  a2)      (»—  o,)(2—  a,)^—  aj 

is  immediately  derived,  provided  that  the  a{  are  so  distributed  as 
to  fulfill  proper  conditions  for  the  convergence  of  the  series.     If 
now  we  take  successively  1,  2,  3,  •  •  •  terms  of  the  expansion,  we 
obtain  the  series  of  polynomials, 


and  it  is  evident  that  Nn(x)  for  the  n  +  1  values  x  =  av  a2,  -  -  •,  an+ 
agrees  in  value  with  l/(z—  x).     By  applying   to  (1)  the  well- 
known  formula  of  Euler  [1,  a]*  for  converting  any  infinite  series 
into  a  continuous  fraction  it  follows  immediately  that  these  poly 
nomials  are  the  successive  convergents  of  the  continued  fraction 


—  CL 


The  generalization  of  formula  (1)  can  be  made  at  once  in  the 
familiar  manner  by  the  use  of  Cauchy's  integral.     We  get  thus 

^  1  f/(z)&  /•^+(g!~a')  r    f(*)dz    -i 

'W  -  2^  J   ^Z^  =/(°.)  -I    -25T~  J   (T~a,)  (z  -  «2)  4 
which  by  placing 

*«(«)  =  (*  -  «i)  (»  -  «2)  ' ' '  (x  ~  O 
may  be  written 

*  Cf.  Encyklopddie  der  Math.  Wiss.,  I  A  3,  p.  134,  formula  (104). 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     157 


For  most  interesting  discussions  of  the  convergence  and  properties 
of  series  having  the  form 

A0  +  A^x  -  aj  +  A2(x  -  a^(x  -  «2)  +  .  .  . 

I  may  refer  to  memoirs  by  Frobenius  [95]  and  Bendixson  [99,  c]. 
I  shall  content  myself  here  with  pointing  out  one  simple  appli 
cation  which  is  given  implicitly  by  both  writers  but  has  been 
noted  again  recently  by  Laurent  [103]. 

Lety(.r)  be  any  analytic  function  the  values  of  which  are  given 
at  a  series  of  points  pi  having  a  regular  point  P  as  their  limit. 
Describe  about  P  as  center  any  circle  C  within  and  upon  which 
f(x)  is  holomorphic,  and  denote  the  points  p.  which  fall  within 
this  circle  by  av  «2,  •  •  •.  Then  lim  a.  =  P.  If  now  z  describes 
the  perimeter  of  the  circle  and  x  is  a  fixed  interior  point,  the 
series  (1)  will  be  uniformly  convergent  and  consequently  permit 
of  integration  term  by  term.  Equation  (2)  therefore  gives  an 
expression  for  f(x)  which  is  valid  in  the  interior  of  C.  This  ex 
pression  shows  at  once  that  an  analytic  function  is  determined 
uniquely  when  its  values  are  known  in  a  sequence  of  points  having 
a  regular  point  P  as  their  limit.  If,  in  particular,  each  /(a.)  =  0, 
f(x)  must  vanish  identically.  In  other  words,  the  zeros  of  an 
analytic  function  can  not  be  infinitely  dense  in  the  vicinity  of  a 
non-singular  point.  Further,  Bendixson  points  out  that  the  con 
vergence  of  the  right  hand  member  of  (2)  is  not  only  the  necessary 
but  the  sufficient  condition  that  /(aj,  /(«2),  /(«3)>  •  •  •  shall  be  the 
values  of  some  analytic  function  at  a  set  of  points  a  .  having  a  limit 
point  P. 

We  turn  now  to  the  generalization  of  the  algorithm  of  the  con 
tinued  fraction.  The  first  investigation  on  this  subject  is  found  in 


158  THE   BOSTON   COLLOQUIUM. 

a  paper  of  Jacobi,*  published  posthumously  in  1868.  The  devel 
opments  of  Jacobi  were,  however,  of  a  purely  numerical  nature. 
On  this  side  they  have  been  perfected  recently  by  Fr.  Meyer  [83] . 
The  first  example  of  a  functional  extension  was  given  by  Hermite 
in  his  famous  memoir  [84]  upon  the  transcendence  of  e,  and  the 
theory  has  been  developed  since  independently  of  each  other  by 
Pincherle  and  Fade. 

To  explain  the  nature  of  the  generalization  it  will  be  desirable 
first  tb  refer  to  the  mode  in  which  a  continued  fraction  is  com 
monly  generated.  Two  numbers  or  functions,  fQ  and/1?  are  given, 
from  which  a  sequence  of  other  numbers  or  functions  is  obtained 
by  placing 

ft  ~  \f\  "~/o> 


/4  =  X3/3  —  f& 

in  which  the  \  are  determined  in  accordance  with  some  stated 
law.     For  the  quotient  fQ/fv  we  obtain  successively 


and  it  therefore  gives  rise  to  the  continued  fraction 

>.-U--v 

By  means  of  the  equations  (3)  each  fn+l  can  be  expressed  linearly 
in  terms  of  the  initial  quantities',/^.     Thus 


in  which  A0tH+l,  A1>n+l  are  polynomials  in  the  elements  X..     It  is 
easy  to  see  that  these  polynomials  both  satisfy  the  same  difference 

*  "  Allgeraeine  Theorie  der  kettenbruchahnlichen  Algorithmen,  in  welchen 
jede  Zahl  aus  drei  vorhergehenden  gebildet  wird."  Journ.  fur  Math.,  vol.  69 
(1868),  p.  29. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     159 
equation  as  fc 

/„+,-*„/.  -/.-,; 

and  for  their  initial  values  we  have 


Consequently  ^4,  „  and  —  AQ>n  are  the  numerator  and  denominator 
of  the  (n  —  l)th  convergent  of  (4). 

When  the  generating  relations  have  the  form 


the  resultant  continued  fraction  is 


A  distinction  then  appears  between  the  system  of  functions 
(Altn+i,  —  -40jW+1)  and  the  system  which  consists  of  the  numerator 
and  denominator  of  the  nth  convergent.  Though  the  quotient 
of  the  two  functions  of  either  system  is  the  ?*th  convergent,  the 
former  pair  of  functions  satisfy  the  same  relation  of  recurrence  as 
the/.,  namely, 

fn  =  \+lfn+l  +  f*n+*f*+*  5 

while  the  corresponding  relation  for  the  other  system  is 

9n  =  \ffn-i  +  P»9n-r 

The  latter  equation  is  called  by  Pincherle  [77,  a]  the  inverse  of 
the  former.  In  the  continued  fraction  (4)  we  took  /z  .  =  —  1  so 
that  the  two  relations  were  coincident. 

The  immediate  generalization  of  these  considerations  is  obtained 
by  taking  m  +  1  initial  quantities  fQ,fiy  •  -  ;fm  in  place  of  two. 
With  a  very  slight  change  of  notation  we  may  write 


160  THE   BOSTON   COLLOQUIUM. 

/.  +  \/,  ++£+'•'*  "mfm  -/„+„ 
(6)  /,  +  X2/2  +  M3/3  +  •  •  •  +  "m+1/m+1  =f 


Jn-m     i~   \-m+iJn-m+l   T  /*n-m+2./n--»i+2      '      '  '  *     •"   VnJn=1  Jn+l* 

Then  by  expressing  /^  in  terms  of  the  m  -f  1  given  quantities  we 
have 


(7) 


m>  nfm, 


in  which  ^  n  is  a  polynomial  in  terms  of  the  Xt.,  /*m,  •  •  • ,  vi+m_l 
(i  =  1,  2,  • .  • ,  n  —  m).  These  m  4-  1  polynomials  ^  n  satisfy 
the  same  difference  equation  (6)  as  the  fn,  and  for  their  initial 
values  we  plainly  have 


•    A 


w=  0         1 
w=  1         0 


m,  n 

o, 
o, 


n  —  m 


0         0 


1. 


Hence  they  constitute  a  complete  system  of  independent  integrals 
of  (6).  Furthermore,  in  analogy  with  the  relation  between  two 
successive  convergents  of  (4), 


=  1, 


we  have  [83,  a,  p.  170] 

A  A 

-^-0,  n  -^1, 

J 


(8) 


A 

0,  n+l    -^l. 


I  ,4  .  .  .    A 

L0,  n+m          1,  n+m  ^  m,  n+m 

The  relation  which  is  the  inverse  of  (6)  has  the  form 


To  obtain  a  system  of  independent  integrals  of  this  equation,  let 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     161 

P0  n  denote  the  minor  of  AQ  n  in  (8),  P,  „  the  minor  of  AltH  after 
the  first  column  has  been  moved  over  the  remaining  columns  so 
as  to  become  the  last,  P2>n  the  minor  of  A2 >M  after  the  first  two 
columns  have  been  moved  over  the  remaining  columns  so  as  to 
become  the  last  two,  and  so  on.  It  can  be  demonstrated  easily 
that  the  desired  system  is  obtained  by  placing  gin+m  =  Pitn 
(i  =  0,  1,  ••-,?/*),  and  these  new  polynomials  rather  than  the 
Ai  n  are  the  true  analogues  of  the  numerator  and  denominator  of 
an  ordinary  continued  fraction.  The  connection  between  the  two 
systems  of  polynomials  is,  however,  both  an  intimate  and  a  re 
ciprocal  one,  for  not  only  is  (9)  the  inverse  of  (6)  but  the  converse 
is  also  true.  On  this  account  the  two  systems  can  be  employed 
simultaneously  with  advantage  in  working  with  the  generalized 
continued  fraction. 

For  all  except  the  very  lowest  values  of  n  the  new  polynomials 
can  be  found  from  the  equations  * 

(»')       P<,n  +  W,  +  ftA»-C  +    '  '  '  +  ".A,-.  =  -P,', .-.-I- 

In  place  of  these  relations  it  will  be  often  found  convenient  to 
employ  such  a  process  as  is  indicated  in  the  following  equations 
for  m  =  2  [83,  a,  p.  180].f 

1 

p      —  9i,  i  >        n^~—  (ft,  i  +  ;.  '—9        n~~=  <?i.  i  +  "  > 


A  4 


?2,2    + 

-  =  9i,i  + 


*Cf.  [83,  a,  p.  174,  eq.  X]. 

fCf.  E.  Fiirstenau,  "  Ueber  Kettenbriiche  hoherer  Ordnung"  ;  Jahresbericht 
ilber  d  is  konigliche  Reakfywuuuimm  zn    \\'ie*ba  ten  ;   1873  4.     See  also  Scott's  De 
terminant?,  Chap.  13,  I  11-12. 
11 


162  THE   BOSTON   COLLOQUIUM. 

We  may  therefore  very  properly  call  the  system  of  values 

X,     it,     •  •  •     i 


\  ^ 


the  norm  of  a  generalized  continued  fraction,  which  itself  consists 
of  the  computation  of  the  Pt  n  or  their  ratios. 

To  apply  this  generalization  to  the  construction  of  algebraic 
continued  fractions,  it  is  only  necessary  to  select  as  the  m  -J- 1  initial 
functions  /0,  •  •  • ,  fm  series  in  ascending  powers  or  series  in 
descending  powers  of  x.  The  nature  of  the  ensuing  theory  will 
be  explained  sufficiently  by  developing  here  the  simplest  case,  in 
which  three  such  series  are  given  [77,  c.]  Take  then 


(*.  *  0), 


S, 


If  we  next  place 

(10)  SQ  +  (aQx  -f-  #o)$!  +  fy)^2  ~  ^3> 

the  coefficients  «0,  a'Q,  60  can  be  so  determined  that  S3  shall  begin 
with  at  least  as  high  a  power  of  \jx  as  the  third.  Normally  the 
degree  is  exactly  3,  and  similarly  for  each  consecutive  value  of  n 
we  have 


in  which  8n  denotes  a  series  beginning  with  the  nth  power  of 
1/x.  Hence  unless  certain  specified  conditions  are  satisfied,  a 
regular  continued  fraction  will  be  obtained  having  the  norm  : 


'  1     <V&  +  a'0 

ft. 

1          n         _1_   n  ' 

6i 

1     a2x  -\-  a'2 

b, 

DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    163 

This  norm  will  not  be  altered  in  any  way  by  dividing  (10) 
through  by  SQ.  It  is  therefore  determined  uniquely  by  the  ratios 
of  £0,  S19  82,  and  conversely  the  ratios  by  the  norm. 

Without  loss  of  generality  we  may  set  S0  =  1.     Place  also 


n+l 


C_    A. 


e.+& 


R 


If  then  n  -f  3  in  (11),  is  replaced  successively  by  n  and  71  +  1, 
and  the  two  equations  are  solved  for  Sl  and  S#  we  obtain 


or 
(12) 
and 
(13) 


Rn 
p 


.fa 


An  examination  of  Pt,  Qn,  Rnt  Xn,  /*n  will  show  that  their  degrees 
in  .r  are 

n— 1,  ?2-2,  H  — 3,   -/'-I,   — r.  (n  =  2r), 

w-1,  n-2,   »-3,   -/'-I,   -/--I  (??  =  9,,+  i). 

Hence  the  expansions  of  Qn  Pn  and  RjPn  in  descending  powers  of 
a?,  agree  with  £,  and  ^2  to  terms  of  degree  3/-  -  1  and  3r  —  2  in 
clusive  if  7i=2r,  and  of  the  3/-th  degree  if  ?i  =  2-r+l.  The 
generalized  continued  fraction  therefore  affords  a  solution  of  the 
problem :  to  find  two  rational  fractions  with  a  common  denom 
inator  which  shall  give  as  close  an  approximation  to  the  given 
functions  £,  and  S2  as  is  consistent  with  the  degrees  prescribed  for 
their  numerators  and  denominators. 

When  three  series  in  ascending  powers  of  x, 


-f 


('"=1,2,3), 


164  THE   BOSTON   COLLOQUIUM. 

are  chosen  as  the  initial  functions,  a  more  comprehensive  algorithm 
can  be  introduced.  Fade  [79,  a]  takes  three  polynomials  A(p}(x\ 
Affi(x),  A(fi(x)  with  undetermined  coefficients,  the  degrees  of  which 
are  indicated  by  their  subscripts,  and  requires  that  their  coefficients 
shall  be  so  determined  that  the  expansion  of 


in  ascending  powers  of  x  shall  begin  with  as  high  a  power  as 
possible.  Ordinarily  this  is  the  (p  +  p  +  p"  +  2)th  power.  To 
each  set  of  values  of  p,  p',  p"  he  shows  that  there  corresponds 
uniquely  a  group  of  three  polynomials  which  he  terms  the  "  asso 
ciated  polynomials,"  and  these  groups  he  arranges  into  a  table  of 
triple  entry  according  to  the  values  of  p,  p'  ,  p".  An  exactly 
similar  table  can  not  be  constructed  for  three  series  in  descend 
ing  powers  of  x,  inasmuch  as  the  substitution  of  1/x  for  x  in 
A^\  •  •  •  i  A&,1  gives  three  rational  fractions,  with  powers  of  x  in 
the  denominators  which  can  not  be  thrown  away  unless 

p=p'  =  p". 

The  new  table  is  handled  by  Fade  in  the  same  manner  as  the 
one  previously  constructed  for  a  single  series.  In  particular,  he 
examines  the  relations 

aA?  +  0A?  +  jAy  =  A?  (i  =  1,  2,  3), 

which  exist  between  four  successive  groups  of  associated  poly 
nomials,  a,  /3,  7  being  rational  functions  of  x  which  are  indepen 
dent  of  the  value  of  i.  When  it  is  possible  to  so  select  a  sequence 
.  •  .  A(*\  A™,  A(r{\  A(*\  A(it\  -  •  •  that  a,  0,  7,  are  polynomials  of 
invariable  degree  for  any  four  consecutive  terms  in  the  sequence, 
the  sequence  or  continued  fraction  is  said  to  be  regular.  In  a 
normal  table  there  are  found  to  be  four  distinct  types  of  such  con 
tinued  fractions.  It  is  worth  noting,  however,  that  the  diagonal 
type  which  was  the  best  in  an  ordinary  table,  no  longer  exists  since 
it  is  found  that  when  the  sequence  fills  a  diagonal  file  of  the  table, 
a,  /3,  and  7  are  no  longer  polynomials  but  rational  fractions  having 
a  common  denominator. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     165 

In  one  important  respect  Pad&s  investigation  has  a  narrower  . 
reach  than  Pincherle's  and  needs  completion.     The  existence  of  a   / 
second  group  of  associated  polynomials  —  the  Pw,  Qn)  Rn  of  Pin- 1 
cherle — is  not  brought  to  light.     As  has  been  already  pointed  out, 
it  is  this  second  group  of  polynomials  which  is  the  true  analogue 
of  the  convergent  of  an  ordinary  continued  fraction  and  which 
must  take  precedence  in  considering  the  convergence  of  the  algo 
rithm  or  the  closeness  of  the  approximation  afforded  to  the  initial 
functions.     Pincherle's  definition  of  convergence  [82]  is  not,  how 
ever,  so  framed  as  to  require  explicitly  the  introduction  of  these 
polynomials.     If  the  given  system  of  difference  equations  is 

(14)  /„„  =  c,/,,+2  +  rf./.+I  +  /„          (n  =  0, 1,  2,  •  •  •), 

the  continued  fraction  is  said  by  him  to  be  convergent  when  the 
two  following  conditions  are  fulfilled  : 

(1)  There  is  a  system  of  integrals  Fn9  Fn,  F^  of  (14)  such  that 
FJFn9  F"JFn  have  limits  for  n  —  oo,  and  these  limits  are  different 
from  0. 

(2)  There  is  also  one  particular  integral  —  called  by  Pincherle 
the  integrate  distinto  —  the  ratio  of  which  to  every  other  integral 
of  (14)  has  the  limit  zero. 

Pincherle's  interest  is  evidently  concentrated  upon  this  prin 
cipal  integral.  It  seems  to  me,  however,  more  natural  to  call 
the  algorithm  convergent  when  the  ratios  Qn/Pn  and  Rn/Pn  (cf. 
Equations  12  and  13)  converge  to  finite  limits  for  n  =  oo.  Under 
ordinary  circumstances  these  limits  will  doubtless  coincide  with 
the  ratios  of  the  generating  functions,  f^f^  audf2/fQ. 

In  the  case  of  an  ordinary  continued  fraction  the  two  definitions 
coalesce.  For  suppose  that  the  ?ith  convergent  NnjDn  of  (4')  has 
the  limit  L.  Then  JV^  —  LDn  is  such  an  integral  of  the  differ 
ence  equation, 

/„  =  \/._,  +  /*./.-„ 

that  its  ratio  to  any  other  integral,  'klNn  -f  £2DM,  has  the  limit  0. 
Conversely,  if  the  principal  integral  Xn  —  LDn  exists,  there  must 
be  a  limit  L  for  the  continued  fraction.  Possibly  the  case  in 


166  THE   BOSTON   COLLOQUIUM. 

which  the  principal  integral  is  Dn  might  be  called  an  excep 
tion,  since  the  continued  fraction  is  then  convergent  by  Pincherle's 
definition,  but  lim  NnjDn  =  oc. 

A  study  of  the  conditions  of  convergence,  so  far  as  I  am  aware, 
has  at  present  been  made  in  only  two  special  cases.  Fr.  Meyer 
[83,  a,  §  7]  has  made  a  partial  investigation  when  the  coefficients 
Xn,  •  •  • ,  vn  in  equations  (6)  are  negative  constants.  Pincherle  [82] 
has  examined  the  case  in  which  the  coefficients  of  the  recurrent 
relation 

/.  +  («.*  +  «:)/.+»  + b,  ,/,«=/n+3 

have  limiting  values  and  finds  that  the  generalized  continued  frac 
tion  is  convergent  for  sufficiently  large  values  of  x.  Let  the  limits 
of  the  coefficients  be  denoted  by  a,  a',  and  b  respectively.  To 
demonstrate  the  convergence  he  avails  himself  of  the  notable  the 
orem  of  JPoincar6,  already  cited  in  Lecture  4.  If,  namely,  no  two 
roots  of  the  equation 

(15)  z3  -  bz2  -  (ax  +  a')/-  1  =  0 

are  of  equal  modulus,  fjfn_l  will  have  a  limit  for  n  —  oo,  and  this 
limit  will  be  one  of  the  roots  of  the  auxiliary  equation  (15), 
usually  the  root  of  greatest  modulus.  From  this  it  follows  di 
rectly  that  AJAn_v  BJBn_v  CJ  Cn_l  as  quotients  of  integrals  of 
the  difference  equation  last  given,  also  PnjPn_v  Qn/  Qn_v  RJRn_l 
as  integrals  of  the  inverse  equation,  have  each  a  definite  limit.  The 
existence  of  limits  for  Qn/Pn  and  of  RnjPn  is  then  established 
for  sufficiently  great  values  of  x,  and  the  analytic  character  of 
these  limits  is  finally  argued.  Let  them  be  denoted  by  U(x)  and 
V(x).  Then  Xn  =  An  +  BnU(x)  +  CnV(x)  is  the  principal  in 
tegral  of  the  difference  equation,  and  has  the  following  distinctive 
property  :  Its  expansion  in  powers  of  1  /x  begins  with  the  highest 
possible  power  consistent  with  the  degrees  of  AH,  Bn,  Cn,  and 
coincides  wiihfn  for  each  successive  value  of  n. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    167 

BIBLIOGRAPHY  OF  MEMOIRS  RELATING  TO  ALGEBRAIC 
CONTINUED  FRACTIONS. 

In  the  following  bibliography  only  works  in  Latin,  Italian, 
French,  German,  and  English  are  included.  In  Wolffin<f&  Mathe- 
matischer  Biicherschatz  (heading  Kettenbruche)  several  dissertations, 
etc.,  are  mentioned  which  may  possibly  relate  to  algebraic  con 
tinued  fractions  but  which  are  not  accessible  to  the  writer.  They 
are  therefore  not  included  here.  The  writer  would  be  glad  to 
have  his  attention  called  to  any  noteworthy  omissions  in  the 
bibliography. 

In  many  cases  it  has  been  extremely  difficult  to  draw  the  line 
between  inclusion  and  exclusion,  especially  under  divisions  vi-ix. 

Any  classification  of  the  -material  which  may  be  adopted  will  be 
open  to  objections,  but  even  an  imperfect  classification  w?ill  prob 
ably  add  greatly  to  the  usefulness  of  the  bibliography.  Since 
much  of  the  work  relating  to  algebraic  continued  fractions  appears 
elsewhere  under  other  headings,  it  is  believed  that  such  a  bibliog 
raphy  as  is  here  given  may  be  of  service. 

For  a  brief  resume"  of  the  theory  of  algebraic  continued  frac 
tions  the  reader  is  referred  to  Osgood's  section  of  the  Encyklopadie 
der  Math.  Wissemchaft,  II  B  i,  §§  38-39. 

I.     ON  THE  DERIVATION  OF  CONTINUED  FRACTIONS  FROM  POWER 
SERIES.     GENERAL  THEORY. 

A.    Early   Works. 

1.  Euler.     (a)  Introductio  in  analysin  infinitorura.  vol.  1,  chap.  18, 

1748. 

(Z>)  De  transformatione  serierum  in  fractiones  contlnuas.  Opus- 
cula  analytica,  vol.  2,  pp.  138-177,  1785. 

2.  Lambert,    (a)  Verwandlimg  der  Briiche.    Beytrage  zum  Gebrauche 

der  Mathematik  und  deren  Anwendung,  vol.  2j,  p.  54  ff.,  p.  161, 
1770. 

(6)  M&moire  sur  quelques  proprietes  remarquables  des  quantit^s 
transcendentes  circulaires  et  logarithmiques.  Histoire  de 
1'Acad.  roy.  des  sciences  et  belles-lettres  &  Berlin,  1768. 


168  THE   BOSTON   COLLOQUIUM. 

3.  Trembley.      Recherches    sur    les    fractions   continues.     Mem.    de 

1'Acad.  roy.  de  Berlin,  1794,  pp.  109-142. 

4.  Kausler.     (a)  Expositio  methodi  series  quascunque  datas  in  frac- 

tiones  continuas  convertendi.     Mem.  de  1'Acad.  imp.  des  sci 
ences  de  St.  Petersbourg,  vol.  1,  pp.  156-174,  1802. 
(b)  De  insigni  usu  fraction um  continuarum  in  calculo  integrale. 
Ibid.,  vol.  1,  pp.  181-194,  1803. 

5.  Viscovatov.     (a)  De  la  method e  generale  pour  reduire  toutes  sortes 

des  quantites  en  fractions  continues.     Ibid.,  vol.  1,  pp.  226-247, 
1805. 

(b)  Essai  d'une  methode  generale  pour  reduire  toutes  sortes  de 
series  en  fractions  continues.  Nova  Acta  Acad.  Scient.  imp. 
Petropolitanaj,  vol.  15,  pp.  181-191,  1802. 

6.  Bret.      Theorie    generale    des    fractions    continues.      Gergonne's 

Annales  de  Math.,  vol.  9,  pp.  45-49,  1818.     Unimportant. 

7.  Scubert.      De    transformatione    seriei    in    fractionem    continuani. 

M6m.  de  1'Acad.  imp.  des  sciences  de  St.  Petersbourg,  vol.  7, 
pp.  139-158,  1820. 

8.  Stern,     (a)  Zur  Theorie  der  Kettenbriiche  und  ihre  Anwendung. 

Jour,  fur  Math.,  vol.  10,  pp.  241-265,  1833. 

(b)  Zur  Theorie  der  Kettenbriiche.  Jour,  fur  Math.,  vol.  18,  pp. 
69-74,  1838. 

9.  Heilermann.     (a)  Ueber  die  Verwandlung  der  Reihen  in  Ketten 

briiche.     Jour,  fur  Math.,  vol.  33,  pp.  174-188,  1846  ;  also  vol. 
46,  pp.  88-95,  1853. 

(b)  Zusammenhang  unter  den  Coefficienten  zweier  gleichen  Ket 
tenbriiche  von  verschiedener  Form.  Zeitschrift  fur  Math,  mid 
Phys.,  vol.  5,  pp.  362-363,  1860.  Unimportant. 

10.  Hankel.     Ueber  die  Transformation  von  Reihen  in  Kettenbriiche. 

Berichte  der  Sachischen  Gesellschaft  der  Wissenschaft  zu  Leip 
zig,  vol.  14,  pp.  17-22,  1862. 

11.  Muir.     (a)  On  the  transformation  of  Gauss'  hypergeometric  series 

into  a  continued  fraction.     Proc.  of  the  London  Math.  Soc., 
vol.  7,  pp.  112-118,  1876. 

(b)  New  general  formulae  for  the  transformation  of  infinite  series 
into  continued  fractions.  Trans,  of  the  R.  Soc.  of  Edinburgh, 
vol.  27,  pp.  467-471,  1876. 

The  general  formulae  in  these  memoirs,  which  Muir  supposed 
to  be  new,  had  been  previously  given  by  Heilermann  in  9(a). 

12.  Heine.     Handbuch  der  Kugelfunction,  2te  Auflage,  1878  ;  chap.  5, 

Die  Kettenbriiche,  pp.  260-297. 

This  gives  a  good  idea  of  the  state  of  the  theory  up  to  1878. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    169 

B.   The  Modern  Theory. 

The  beginnings  of  this  theory  are  to  be  found  in  Nos.  110 
and  111. 

13.  Frobenius.     ITeber  Relationeu  zwischen  den  Naherungsbriichen  von 

Potenzreihen.     Jour.  fur  Math.,  vol.  90,  pp.  1-17,  1881. 

This  fundamental  memoir  marks  an  important  advance.  See 
16(o). 

14.  Stieltjes.     Sur  la  reduction  en  fraction  continue  d'une  serie  pro- 

cedant  suivant  les  puissances  descendantes  d'une  variable. 
Ann.  de  Toulouse,  vol  3,  H,  pp.  1-17,  1889. 

15.  Pincherle.     Sur  une  application  de  la  theorie  des  fractions  contin 

ues  algebriques.     Comp.  Rend.,  vol.  108,  p.  888,  1889. 

16.  Fade,     (a)  Sur  la  representation  approchee  d'une  fonction  par  des 

fractions  rationnelles.  Thesis,  published  in  the  Ann.  de  1'Ec. 
Nor.,  ser.  3,  vol.  9,  supplement,  pp.  1-93,  1892. 

This  very  fundamental  memoir  is  the  best  one  to  read  for  the 
purpose  of  learning  the  elements  of  the  theory  of  algebraic 
continued  fractions.  The  same  point  of  view  is  taken  as  by 
Frobenius  in  (13)  and  is  more  completely  developed.  The 
thesis  was  preceded  by  the  two  following  preliminary  notes  : 

(ax)  Sur  la  representation  approchee  d'une  fonction  par  des 
fractions  rationnelles.  Comp.  Rend,  vol.  Ill,  p.  674,  1890. 

(a")  Sur  les  fractions  continues  regulieres  relatives  a  e*. 
Comp.  Rend,  vol.  112,  p.  712,  1891. 

(ft)  Recherches  nouvelles  sur  la  distribution  des  fractions 
rationnelles  approchees  d'une  fonction.  Ann.  de  1'Ec.  Nor., 
ser.  3,  vol.  19,  pp.  153-189,  1902. 

(c)  Apercu  sur  les  developpements  recents  de  la  theorie  des 
fractions  continues.  Compte  rendu  du  deuxieme  Congres  inter 
national  des  mathe"maticiens  tenu  a  Paris,  pp.  257-264,  1900. 

Only  a  restricted  portion  of  the  field  is  here  reviewed,  and  in 
this  portion  the  important  work  of  Pincherle  is  overlooked. 

17.  Fade,     (a)  Sur  les  series  entieres  convergentes  ou  divergentes  et 

les  fractions  continues  rationelles.  Acta  Math.,  vol.  18,  pp. 
97-111,  1894. 

(a')  Sur  la  possibility  de  definir  une  fonction  par  une  serie 
entiere  divergente.     Comp.  Rend.,  vol.  116,  p.  686,  1893. 
See  also  No.  26a3  76. 

II.     ON  CONVERGENCE. 

(For  a  resume  of  the  criteria  for  the  convergence  of  continued 
fractions  with  real  elements  see  PRINGSHEIM'S  report  in  the  En- 
cyklopadie  der  mathematischen  Wissenschaften,  I  A  3,  p.  126,  ff.) 


170  THE   BOSTON    COLLOQUIUM. 

18.  Riemann.  Sullo  svolgimento  del  quoziente  di  due  serie  ipergeo- 
metriche  in  frazione  continua  infinita,  1863.  Gesammelte  inath- 
ematische  Werke,  pp.  400-406. 

18,  bis.  Worpitzky.     Untersuchung  iiber  die  Entwickelung  der  mono- 

dromen  und  monogenen  Functionen  durch  Kettenbriiche.     Pro- 
gramm,  Friedrichs  Gyinnasiuoi  und  Realschule,  Berlin,  1865. 

This  program  and  the  two  following  memoirs  of  Thome  were 
published  before  Riemann' s  posthumous  fragment. 

19.  Thome,     (a)  Ueber  die  Kettenbruchentwickelung  der  Gauss' schen 

Function  F(a,  1,  y,  x).     Jour,  fur  Math.,  vol.  66,  pp.  322-336, 
1866. 

(b)  Ueber  die  Kettenbruchentwickelung  des   Gauss' schen  Quo- 
tienten 


Ibid.,  vol.  67,  pp.  299-309,  1867. 
20.     Laguerre.     Sur  1' integrate 

C™e~x  i 
\       —  dx. 

Jx          X 


Bull,  de  la  Soc.  Math,  de  France,  vol.  7,  pp.  72-81,  1879,  or 
Oeuvres,  vol.  1,  p.  428. 

Historically  an  important  memoir  because  of  its  development 
of  the  connection  between  a  divergent  power  series  and  con 
vergent  continued  fraction.     See  the  first  footnote  in  lecture  4  ; 
also  No.  102,  p.  30. 
21.  Halphen.     (a)   Sur  la  convergence  d'une  fraction  continue  alge- 

brique.     Comp.  Rend.,  vol.  100  (1885),  pp.  1451-1454. 
(6)  Same  subject.    Ibid.,  vol.  106  (1888),  pp.  1326-1329. 
(c)  Traite  des  fonctions  elliptiques.     Chap.   14.     Fractions   con 
tinues  et  integrates  pseudo-elliptiques. 

.  22.  Pincherle.     Alcuni  teoremi  suite  frazioni  continue.     Atti  delle  R. 
Accad.  dei  Lincei,  ser.  4,  vol.  5X,  pp.  640-643,  1889. 

The  test  for  convergence  given  here  is  included  in   a  more 
general  criterion  given  later  by  Pringsheim,  No.  29. 
.23.  Pincherle.     Sur  les  fractions  continues  algebriques.     Ann.  de  1'Ec. 
Nor.,  ser.  3,  vol.  6,  pp.  145-152,  1889. 

An  incomplete  result  is  here  obtained.    See  No.  32c  for  the 
complete  theorem. 

24.  Fade.     Sur  la  convergence  des  fractions  continues  simples.     Comp. 
Rend.,  vol.  112,  p.  988, 1891.    Also  found  in  §§  45-47  of  No.  16a. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     171 

25.  Banning.     Ueber  Kugel-  und  Cylinderfunktionen  und  deren  Ket- 

tenbruchentwickelung.     Dissertation,  Bonn,  1894,  pp.  1-33. 

26.  Stieltjes.     (a)  Recherches  sur  les  fractions  continues.     Annales  de 

Toulouse,  vol.  8,  J,  pp.  1-122,  and  vol.  9,  A,  pp.  1-47.  1894-95. 
Published  also  in  vol.  32  of  the  Memoires  presenters  a  1'Acad. 
des  sciences  de  1'Institut  National  de  France. 

A  rich  memoir,  developing  particularly  the  connection 
between  an  important  class  of  continued  fractions  and  the  cor 
responding  integrals. 

(a')  Sur  un  developpement  en  fraction  continue.  Comp. 
Rend.,  vol.  99,  p.  508,  1884. 

(a")  Same  subject.     Ibid.,  vol.  108  (1889),  p.  1297. 

(a/7/)  Sur  une  application  des  fractions  continues.  Ibid.,  vol. 
118  (1894),  p.  1315. 

(aTV)  Recherches  sur  les  fractions  continues.  Ibid.,  vol.  118 
(1894),  p.  1401. 

Markoff.  (b)  Note  sur  les  fractions  continues.  Bull,  de  1'Acad. 
imp.  des  sciences  de  St.  Petersbourg,  ser.  5,  vol.  2,  pp.  9-13, 
1895. 

This  gives  a  discussion  of  the  relation  of  his  work  to  that  of 
Stieltjes. 

27.  H.  von  Koch,     (a)  Sur  un  theoreme  de  Stieltjes  et  sur  les  fonctions 

definies  par  des  fractions  continues.  Bull,  de  la  Soc.  Math,  de 
France,  vol.  23,  pp.  33-40,  1895. 

(«')  Sur  la  convergence  des  determinants  d'ordre  infini  et  des 
fractions  continues.  Comp.  Rend.,  vol.  120,  p.  144,  1895. 

28.  Markoff.     Deux  demonstrations  de  la  convergence  de  certaines  frac 

tions  continues.     Acta  Math.,  vol.  19,  pp.  93-104,  1895. 

Contained  also  in  his  Differenzenrechnung  (deutsche  Ueber- 
setzung),  chap.  7,  §  21-22. 

This  discusses  the  convergence  of  the  usual  continued  frac 
tion  for 


z  —  y 

when/(7/)  >  0  between  the  limits  of  integration. 

29.  Pringsheim.  Ueber  die  Convergenz  unendlicher  Kettenbruche. 
Sitzungsberichte  der  math.-phys.  Classe  der  k.  bayer'schen 
Akad.  der  Wissenschaften,  vol.  28,  pp.  295-324,  1898. 

The  most  comprehensive  criteria  for  convergence  yet  obtained 
are  found  in  29,  31,  and  325. 


172  THE    BOSTON   COLLOQUIUM. 

30.  Bortolotti.     Sulla  convergenza  delle  frazioni  continue  algebriche. 

Atti  della  R.  Accad.  del  Lincei,  ser.  5,  vol.  8,,  pp.  28-33,  1899. 

31.  Van  Vleck.     On  the  convergence  of  continued  fractions  with  com 

plex  elements.     Trans.  Amer.  Math.  Soc.,  vol.  2,  pp.  215-233, 
1901. 

32.  Van  Vleck.     (a)  On  the  convergence  of  the  continued  fraction  of 

Gauss  and  other  continued  fractions.     Annals  of  Math.,  ser.  2, 
vol.  3,  pp.  1-18,  1901. 
(b)  On  the  convergence  and  character  of  the  continued  fraction 


Trans.  Amer.  Math.  Soc.,  vol.  2,  pp.  476-483,  1901. 
(c)  On  the  convergence  of  algebraic  continued  fractions  whose 
coefficients  have  limiting  values.     Ibid.,  vol.  5,  pp.  253-262, 
1904. 

33.  Montessus.     (a)  Sur  les  fractions  continues  algebriques.      Bull,  de 

la  Soc.  Math,  de  France,  vol.  30,  pp.  28-36,  1902. 

The  content  of  this  memoir  was  discussed  in  lecture  5. 
(b)  Same  title.     Comp.  Rend.,  vol.  134  (1902),  p.  1489. 
See  also  37a',  41. 

III.   ON  VARIOUS  CONTINUED  FRACTIONS  OF  SPECIAL  FORM. 
A.   The  Continued  Fraction  of  Gauss. 

34.  Gauss.     Disquisitiones  generales  circa  seriem  infinitam 


Deutsche  Uebersetzung  von  Simon,  or  Werke,  vol.  3,  pp.  134- 
138,  1812. 

34,  bis.  Vorsselman  de  Herr.     Specimen  inaugurale  de  fractionibus  con- 
tinuis.     Dissertation,  Utrecht,  1833. 

Numerous  references  are  given  here  to  the  early  literature 
upon  continued  fractions. 

34,  ter.  Heine.     Auszug  eines  Schreibens  iiber  Kettenbriiche  von  Herrn 

E.  Heine  an  den  Herausgeber.     Jour,  fiir  Math.,  vol.  53,  pp. 
284-285,  1857. 
See  also  40c,  p.  231. 

35.  Euler.     (a)  Commentatio  in  fraction  em  continuam  in  qua  illustris 

Lagrange  potestates  binomiales  expressit.    Memoires  de  1'Acad. 
imp.  des  sciences  de  St.  Petersbourg,  vol.  6,  pp.  3-11,  1818. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    173 

Fade,  (b)  Sur  la  generalisation  des  deVeloppements  en  fractions 
continues,  donnes  par  Gauss  et  par  Eider,  de  la  fonction 
(1  +  x)m.  Comp.  Rend. ,  vol.  129,  p.  753,  1899. 

(c)  Sur  la  generalisation  des  developpements  en  fractions  contin 
ues,  donnes  par  Lagrange  de  la  fonction  (1  +  x)m-     Ibid.,  vol. 
129,  p.  875,  1899. 

(d)  Sur  1' expression  generale  de  la  fraction  rationnelle  approchee 
de  (1  +  x)m.     Ibid.,  vol.  132,  p.  754,  1901. 

See  also  Nos.  11,  32a,  65. 

B.   The  Continued  Fractions  for  e*. 

36.  Winckler.     Ueber  angenaherte  Bestinimungeu.     Wiener  Berichte, 

Math.-naturw,  Classe,  vol.  72,  pp.  646-652,  1875. 

37.  Fade,     (a)  Meraoire  sur  les  developpements  en  fractions  continues 

de  la  fonction  exponentielle,  pouvant  servir  d' introduction  a  la 
theorie   des   fractions   continues   algebriques.      Ann.   de   1'Ec. 
Nor.,  Ser.  3,  vol.  16,  pp.  395-426,  1899. 
(a')  Sur  la  convergence  des  reduites  de  la  fonction  exponentielle. 

Comp.  Rend.,  vol.  127,  p.  444,  1898. 
See  also  Nos.  16a",  106,  and  pages  243-5  of  40c. 

C.  The  Continued  Fraction  of  Bessel. 

38.  Giinther.     Bemerkungen   iiber   Cylinder-Functionen.     Archiv   der 

Math,  und  Phys.,  vol.  56,  pp.  292-297,  1874. 

39.  Graf,     (a)  Relations  entre  la  fonction  Besselienne  de  1"  espece  et 

une  fraction  continue.    Annali  di  Mat.,  ser.  2,  vol.  23,  pp.  45-65, 
1895. 

Giving  references  to  earlier  works  where  the  continued  frac 
tion  of  Bessel  is  found. 

Crelier.     (b)  Sur   quelques  proprietes   des  fonctions   Besseliennes, 
tirees  de  la  theorie  des  fractions  continues.     Annali  di  Mat., 
vol.  24,  pp.  131-163,  1896. 
See  also  Nos.  25,  32a. 

D.  The  Continued  Fraction  of  Heine. 

40.  Heine,     (a)  Ueber  die  Reihe 

a       l-l)(g6_1)(gp  +1_1) 

1  "• 


) 


174  THE  BOSTON  COLLOQUIUM. 

Jour,  fur  Math.,  vol.  32,  pp.  210-212,  1846. 

(b)  Untersuchung  iiber  die  (selbe)  Reihe.    Ibid.,  vol.  34,  pp.  285- 
328,  1847. 

(c)  Ueber  die  Zahler  und  Nenner  der  Naherungswerthe  von  Ket- 
tenbriiche.     Ibid.,  vol.  57,  pp.  231-247,  1860. 

Christoffel  (d)  Zur    Abhandlung    " Ueber    Zahler    und   Nenner" 
(u.  s.  w.)  des  vorigen  Bandes.     Ibid.,  vol.  58,  pp.  90-91,  1861. 

41.  Thomae.     Beitriige  zur  Theorie  der  durch  die  Heine' sche  Reihe 

darstellbaren  Funktionen.    Jour,  fur  Math.,  vol.  70,  1869.    See 
pp.  278-281  where  the  convergence  of  Heine's  continued  frac 
tion  is  proved. 
See  also  32a. 

42.  (On  Eisensteiri's  continued  fractions). 

Heine,     (a)  Verwandlung  von  Reihen  in  Kettenbriiche.     Jour,  fur 
Math.,  vol.  32,  pp.  205-209,  1846. 

See  also  vol.  34,  p.  296. 

Muir.     (b)  On  Eisenstein's  continued  fractions.     Trans.  Roy.  Soc. 
of  Edinburgh,  vol.  28,  part  1,  pp.  135-143,  1877. 

Muir  plainly  was  not  aware   of  the   preceding  memoir   by 
Heine. 

E.   The  Continued  Fraction  of  Stieltjes.     (See  No.  26.) 

43.  Borel.     Les  series  de   Stieltjes,    Chap.  5  of  his  Memoire  sur    les 

series divergentes.  Ann.  del'Ec.  Nor.,  ser.  3,  vol.  16,  pp.  107- 
128  ;  and  also  chap.  2  of  his  treatise,  Les  Series  divergentes, 
pp.  55-86,  1901. 

44.  Fade.   Sur  la  fraction  continue  de  Stieltjes.    Comp.  Rend.,  vol.  132, 

p.  911,  1901. 

45.  Van^Vleck.      On   an   extension   of  the   1894   memoir   of  Stieltjes. 

Trans.  Amer.  Math.  Soc.,  vol.  4,  pp.  297-332,  1903. 
See  also  Nos.  27,  102. 

F.   The  Continued  Fraction  for 
1  +  mx  +  m(m  +  7i)x2  +  m(m  +  n)  (m  -f  2ri)x*  -\ 

and  its  special  cases. 

46.  Euler.     (a)  De  seriebus  divergentibus.     Novi  commentarii  Acad. 

scientiarum  iinperialis  Petropolitanse,  vol.  5,  pp.  205-237,  1754- 
5  ;  in  particular  pp.  225  and  232-237. 
(b)  De  transformatione  seriei  divergentis 

1  —  mx  -f  m(m  -f-  ri)x"*  —  m(m  +  n)  (m  +  2ri)z?  -f  •  •  • 

I 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     175 

in  fractioneni   coutinuam.     Nova  acta  Acad.  scientiarum  im- 
perialis  Petropolitanse,  vol.  2,  pp.  36-45,  1784. 
Gergonne.     (c)  Recherches  sur  les  fractions  continues.     Gergonne's 
Annales  de  Math.,  vol.  9,  pp.  261-270,  1818. 

47.  Laplace,      (a)  Trait6  de  mecanique  celeste.     Oeuvres,  vol.  4,  pp. 

254-257,  1805. 

/»»  -x* 
Jacobi.     (b)   De   fractione   continua   in   quam   integrale  I     e    dx 

•Jx, 

evolvere  licet.  Jour,  fur  Math.,  vol.  12.  pp.  346-347,  1834,  or 
Werke,  vol.  6,  p.  76. 

See  also  p.  79  of  No.  20,  and  the  first  note  under  lecture  2. 

G.  Periodic  Continued  Fractions,  and  Continued  Fractions  Connected  with 
the  Theory  of  Elliptic  functions. 

48.  Abel,     (a)  Sur  V integration  de  la  formule  differentielle  pdxlVR,  R 

et  p  etant  des  fonctions  entieres.  Jour,  fur  Math.,  vol.  1,  pp. 
185-221,  1826,  or  Oeuvres,  vol  1,  p.  104  if. 

Dobma.     (b)  Sur  le  developpement  de  VR  en  fraction  continue. 
Nouvelles  Ann.  de  Math.,  ser.  3,  vol.  10,  pp.  134-140,  1891. 

49.  Jacobi.     (a)  Note  sur  une  nouvelle  application   de  1' analyse   des 

fonctions  elliptiques  a  1'algebre.  Jour,  fur  Math.,  vol.  7,  pp. 
41-43,  1831,  or  Werke,  vol.  1,  p.  327. 

Borchardt.     (6)    Application    des   transcendantes   abeliennes   a   la 
theorie  des  fractions  continues.     Ibid.,  vol.  48,  pp.  69-104,  1854. 

50.  Tchebychef.     Sur  F  integration  des  differentielles  qui   contiennent 

une  racine  carree  d'un  polynome  du  troisieme  ou  du  quatrieme 
degre.  Memoires  de  FAcad.  imp.  des  sciences  de  St.  Peters- 
bourg,  ser.  6,  vol.  8,  pp.  203-232,  1857. 

51.  Frobenius  und  Stickelberger.     Ueber  die  Addition  und  Multiplication 

der  elliptischen  Functionen.  Jour,  fur  Math.,  vol.  88,  pp.  146- 
184,  1880. 

52.  Halphen.     Sur  les  integrates  pseudo-elliptiques.     Comp.  Rend.,  vol. 

106  (1888),  pp.  1263-1270. 

53.  Bortolotti.     Sulle  frazioni  continue  algebriche  periodiche.     Rendi- 

conti  del  Circolo  Mat.  di  Palermo,  vol.  9,  pp.  136-149,  1895. 
See  also  Nos.  21,  26(a),  40. 

H.  Miscellaneous. 

54.  Euler.     (a)  Speculations  super  formula  integrali 

xndx 


/a2  —  2bx  +  ex2 


176  THE  BOSTON   COLLOQUIUM. 

ubi  simul  egregiae  observationes  circa  fractiones  continuas  occur- 
rent.  Acta  Acad.  scientdarum  imperialis  Petropolitanae,  1784, 
pars  posterior,  pp.  62-84,  1782. 

(b)  Summatio   fractionis   continuse    cujus  indices   progression  em 
arithmeticam  constitimnt.    Opuscula  Analytica,  vol.  2,  pp.  217- 
239,  1785. 
55.  Spitzer.     (a)  Darstellung  des  unendlichen  Kettenbruchs 


in  geschlossener  Form,  nebst  anderen  Bemerkungen.     Archiv 
der  Math,  und  Phys.,  vol.  30,  pp.  81-82,  1858. 
(6)  Darstellung  des  unendlichen  Kettenbruchs 


in  geschlossener  Form.     Ibid.,  vol.  30,  pp.  331-334,  1858. 

(c)  Note  iiber  eine  Kettenbriiche.     Ibid.,  vol.  33,  pp.  418-420, 
1859. 

(d)  Darstellung  des  unendlichen  Kettenbruches 


1   n(2x  +  3)  +  n(2x  +  5)  + 

in  geschlossener  Form.     Ibid.,  vol.  33,  pp.  474-475,  1859. 
56.  Laurent,     (a)  Note  sur  les  fractions  continues.     Nouvelles  Ann.  de 
Math.,  ser.  2,  vol.  5,  pp.  540-552,  1866. 
This  treats  the  continued  fraction 


1+1+1+ 
E.   Meyer,      (b)     Ueber     eine     Eigenschaft     des     Kettenbruches 

x —      •  •  • .    Archiv  der  Math,  und  Phvs. ,  ser.  3,  vol.  5, 

x  —  x  — 

p.  287,  1903. 

Meyer's  results  will  be  found  on  p.  548  of  Laurent's  memoir 
and  differs  only  in  that  x  has  been  replaced  by  —  l/x2. 
57.  Schlomilch.     (a)  Ueber  den  Kettenbruch  fur  tan  z.    Zeitschrift  fur 

Math,  und  Phys.,  vol.  16,  pp.  259-260,  1871. 
Glaisher.     (b)   A   continued   fraction   for  tan   nx.      Messenger   of 

Math.,  ser.  2,  vol.  3,  p.  137,  1874. 

(c)  Note  on  continued  fractions  for  tan  nx.     Ibid.,  ser.  2,  vol.  4, 
pp.  65-58,  1875. 


". 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     177 

j  58.  Schlomilch.  Ueber  die  Kettenbruchentwickelung  fur  unvollstan- 
dige  Gamma-function.  Zeitschrift  fur  Math,  und  Phys.,  vol. 
16,  pp.  261-262,  1871. 

This  gives  the  development  of  I     t*~l  e~fdt. 

•Jo 

59.  Schendel.     Ueber  eine  Kettenbruchentwickelung.    Jour,  fur  Math., 

vol.  80,  pp.  95-96,  1875. 

60.  Lerch.     Note  sur  les  expressions  qui,  dans  diverses  parties  du  plan, 

representent  des  fonctions  distinctes.     Bull,  des  sciences  Math, 
ser.  2,  vol.  10,  pp.  45-49,  1886. 

61.  Stieltjes.    (a)  Sur  quelques  integrates  definies  et  leur  developpement 

en  fractions  continues.    Quar.  Jour,  of  pure  and  applied  Math., 
vol.  24,  pp.  370-382,  1890. 

(£>)  Note  sur  quelques  fractions  continues.  Ibid.,  vol.  25,  pp.  198- 
200,  1891. 

62.  Hermite.     Sur  les  polynomes  de  Legendre.     Jour,  fur  Math.,  vol. 

107,  pp.  80-83,  1891. 

This  connects  DWl*n\x)  with  a  continued  fraction. 

IV.  ON  THE  CONNECTION  OF  CONTINUED  FRACTIONS  WITH  DIFFEREN 
TIAL  EQUATIONS  AND  INTEGRALS. 

A.  EiccatVs  Differential  Equation. 

63.  Euler.     (a)  De  fractionibus  continuis  observation es.     Commentarii 

academise   scientiarum   imperialis   Petropolitanae,  vol.  11,  see 
pp.  79-81,  1739. 

(6)  Analysis  facilis  sequationem  Riccatianam  per  fractionem  con- 
tinuam  resolvendi.  Memoires  de  1'  Acad.  imperiale  des  sciences 
de  St.  Petersbourg,  vol.  6,  pp.  12-29,  1813. 

64.  Lagrange.     Sur  1'usage  des  fractions  continues  dans  le  calcul  inte*- 

gral.     Nouveaux  Mem.  de  1'Acad.  roy.  des  sciences  et  belles- 
lettres  de  Berlin,  1776,  pp.  236-264,  or  Oeuvres,  vol.  4,  p.  301  flf. 

One  of  the  few  important  early  works. 
See  546  ;  also  No.  66a  for  work  on  differential  equations  of  the  1st  order. 

B.    Miscellaneous  Differential  Equations  of  the  Second  Order. 

In  a  numerous  class  of  continued  fractions  the  denominators 
of  the  convergents  satisfy  allied  (Heun,  "  gleichgruppige  ")  differ 
ential  equations  of  the  second  order.  Early  instances  are  found 
in  works  of  Gauss  (No.  114),  Jacobi  (No.  65)  and  Heine  (No.  72). 
The  theory,  from  two  different  aspects,  is  furthest  developed  in 
66a  and  76. 


178  THE   BOSTON   COLLOQUIUM. 

65.  Jacobi.     Untersuchung  iiber  die  Differentialgleichung  der  hyper- 

geometrischen  Reihe.  Nachlass.  Jour,  fur  Math.,  vol.  56, 1859  ; 
see  in  particular  §  8,  pp.  160-161,  or  Werke,  vol.  6,  p.  184. 

66.  Laguerre.     (a)  Sur  la  reduction  en  fractions  continues  d'une  frac 

tion  qui  satisfait  a  une  equation  differentielle  lineaire  du  pre 
mier  ordre  dont  les  coefficients  sont  rationnels.  Jour,  de  Math., 
ser.  4,  vol.  1,  pp.  135-165,  1885.  (jt**^  V*l  y.  ^\^5 

(This  is  a  comprehensive  memoir  which  incorporates  substan 
tially  all  the  following  memoirs  : 

(b)  Sur  la  reduction  en   fractions   continues   d'une   classe   assez 
>>     6tendue  de  fonctions.     Comp.  Rend.,  vol.  87  (1878),  p.  923,  or 

Oeuvres,  vol.  1,  p.  322. 
>.      (c)  Same  title  as  (a).     Bull,  de  la  Soc.  Math,  de  France,  vol.  8 

(1880),  pp.  21-27,  or  Oeuvres,  vol.  1,  p.  438. 
(d)  Sur  la  reduction  en  fraction  continue  d'une  fraction  qui  satis 
fait  a  une  equation  lineaire  du  premier  ordre  a  coefficients  ration 
nels.     Comp.  Rend.,  vol.  98  (1884),  pp.  209-212  or  Oeuvres, 
vol.  1,  p.  445. 

67.  Laguerre.     (a)  Sur   1' approximation   des  fonctions  d'une   variable 

au  moyen  de  fractions  rationnelles.     Bull,  de  la  Soc.  Math,  de 
France,  vol.  5  (1877),  pp.  78-92  or  Oeuvres,  vol.  1,  p.  277. 
(b)  Sur  le  developpement  en  fraction  continue  de 


Ibid.,  vol.  5  (1877),  pp.  95-99  or  Oeuvres,  vol.  1,  p.  291. 

(x  -f-  1  \  w 
*  )  • 
x —  1  / 

Ibid.,  vol.  8  (1879),  pp.  36-52,  or  Oeuvres,  vol.  1,  p.  345. 

(d)  Sur  la  reduction  en  fractions  continues  de  eF(x>,  F(x)  desig- 
nant  un  polyn6me  entier.  Jour,  de  Math.,  ser.  3,  vol.  6  (1880), 
pp.  99-110,  or  Oeuvres,  vol.  1,  p.  325. 

(d')  Same  subject.  Comp.  Rend. ,  vol.  87  (1878),  p.  820,  or  Oeuvres, 
vol.  1,  p.  318. 

68.  Humbert.     Sur  la  reduction  en  fractions  continues  d'une  classe  de 

fonctions.     Bull,  de  la  Soc.  Math,  de  France,  vol.  8,  pp.  182- 
187,  1879-1880. 

69.  Hermite  et  Fuchs.     Sur   un   developpement  en  fraction  continue. 

Acta  Math.,  vol.  4,  pp.  89-92,  1884. 
See  also  No.  20,  34  ter,  71-76. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     179 

C.   Differential  Equations  of  Order  Higher  than  the  Second. 

70.  Pincherle.     Sur   la   generation   de   systemes  recurrents  au  moyen 

d'une  equation  lineaire  differentielle.     Acta  Math.,  vol.  16,  pp. 
341-363,  1892-3. 
See  also  No.  15,  86,  87,  1246. 

D.    The  integral  CfW*' 

Ja      X  —  z 

71.  Heine,     (a)  Ueber  Kettenbriiche.     Monatsberichte  der  k.  preussi- 

schen  Akad.  der  Wissenschaften  zu  Berlin,  1866,  pp.  436-451. 
(a')  Mittheilung  iiber  Kettenbruche.    Auszug  aus  dem  Monatsbe 
richte,  u.  s.  w.     Jour,  fur  Math.,  vol.  67,  pp.  315-326,  1867. 
See  also  Nos.  12,  26a,  28,  45,  102,  113,  118a. 

E.  HypereUiptic  and  Similar  Abelian  Integrals. 

72.  Heine.     Die   Lame'schen    Functionen  verschiedener   Ordnungen. 

Jour,  far  Math.,  vol.  60,  1862,  pp.  252-303  ;  in  particular  pp. 
256,  275,  294-297.  Or  see  his  Handbuch,  vol.  1  (2te  Auf.),  pp. 
388-396  and  468. 

73.  Laguerre.     Sur  1' approximation  d'une  classe  de  transcendantes  qui 

comprennent  comme  cas  particulier  les  integrates  hyperellip- 
tiques.     Comp.  Rend.,  vol.  84,  pp.  643-645,  1877. 
(Not  found  in  vol.  1.  of  his  Oeuvres.) 

74.  Humbert.     Sur  1' equation  difterentielle  lineaire  du  second  ordre. 

Jour,  de  1'Ec.  Polytech.,  vol.  29,  cahier  48,  pp.  207-220,  1880. 

75.  Heun.     (a)  Die  Kugelfunctionen  und  Lame'schen  Functionen  als 

Determinanten.     Dissertation,  pp.  1-32,  Gottingen.  1881. 
(6)  Ueber  lineiire  Differentialgleichungen  zweiter  Ordnung  deren 
Losungen  durch  den  Kettenbruchalgorithmus  verknupft  sind. 
Habilitationsschrift.     1881. 

(c)  Integration  regularer  linearer  Differential gleichungen  zweiter 
Ordnung  durch  die  Kettenbruchentwickelting  von  ganzen  Abel'- 
schen  Integralen  dritter   Gattung.     Math.  Ann.,  vol.  30,  pp. 
553-560,  1887. 

(d)  Beitriige   zur   Theorie   der   Lame'schen   Functionen.     Math. 
Ann.,  vol.  33,  pp.  180-196,  1889. 

The  important  group-properties  of  the  continued  fraction  are 
here  brought  out  and  are  further  developed  in  No.  76. 

76.  Van  Vleck.     Zur  Kettenbruchentwickelung  hyperelliptischer  und 

ahnlicher  Integrale.  Dissertation,  Gottingen  ;  published  in  the 
Amer.  Jour,  of  Math.,  vol.  16  (1894),  pp.  1-91. 


180  THE   BOSTON   COLLOQUIUM. 

After  development  first  from  an  algebraic  standpoint  the  sub 
ject  is  carried  further  by  the  method  of  con  formal  representation. 
The  suggestion  of  this  treatment  is  given  in  Klein's  Differen 
tial  gleichungen,  1890-91,  vol.  1,  pp.  180-186. 

V.    GENERALIZATION  OF  THE  ALGEBRAIC  CONTINUED  FRACTION. 
A.   General  Theory. 

So  far  as  I  have  been  able  to  ascertain,  the  first  instance  of  the 
generalization  is  contained  in  Hermite's  memoir,  No.  84.  The 
development  of  a  general  theory  is  due  to  Fade  and  Pincherle. 
Nos.  77a,  776,  and  79a  are  especially  recommended. 

*_  77.  Pincherle.  (a)  Saggio  di  una  generallizzazione  delle  frazioni  con 
tinue  algebriche.  Memoirie  della  R.  Accad.  delle  Scienzre  dell' 
Istituto  di  Bologna,  ser.  4,  vol.  10,  p.  513-538,  1890. 
(a7)  Di  un'estensione  dell'  algorithmo  delle  frazioni  continue. 
Kendiconti,  R.  Istituto  Lombardo  di  Scienze  e  Lettere,  ser.  2, 
vol.  22,  pp.  555-558,  1889. 

(b)  Sulla  generalizzazione  delle  frazioni  continue  algebrique.     An- 
nali  di  Mat.,  ser.  2,  vol.  19,  pp.  75-95,  1891. 

78.  Hermite.    Sur  la  generalisation  des  fractions  continues  algebriques. 

Annali  di  Mat.,  ser.  2,  vol.  21,  pp.  289-308,  1893. 

79.  Fade,     (a)   Sur    la    generalisation    des   fractions    continues   alge 

briques.     Jour,  de  Math.,  ser.  4,  vol.  10,  pp.  291-329,  1894. 
(O  Same  subject.     Comp.  Kend.,  vol.  118,  p.  848,  1894. 

80.  Bortolotti.     Un  contribute  alia  teoria  delle  forme  lineari  alle  differ- 

enze.     Annali  di  Mat.,  ser.  2,  vol.  23,  pp.  309-344,  1895. 

81.  Cordone.     Sopra  un  problema  fundamental  delle  teoria  delle  fra 

zioni  continue  algebriche  generalizzate.     Rendiconti  del  Circolo 
di  Palermo,  vol.  12,  pp.  240-257,  1898. 

Cordone  seeks  the  regular  algorithms  which  are  similar  to 
those  of  Fade  but  occur  in  connection  with  n  series  in  descend 
ing  powers  of  x. 

B.   Convergence  of  the  Generalized  Algorithm. 

82.  Pincherle.     Contribute  alia  generalizzazione  delle  frazioni  continue. 

Memoirie  della  R.  Accad.  delle  Scienze  dell'  Istituto  di  Bologna, 
ser.  5,  vol.  4,  pp.  297-320,  1894. 

83.  W.  Franz  Meyer,     (a)  Ueber   kettenbruchahnlichen   Algorithmen. 

Yerhand.  des  ersten  internationalen  Mathematiker-Kongresses 
in  Zurich,  pp.  168-181,  1898  ;  see  in  particular  §7. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     181 

(a')  Zur  Theorie  der  kettenbruchahnlichen  Algorithmen.  Schrif- 
ten  der  phys-okonomischen  Gesellschaft  zu  Konigsberg,  vol. 
38,  pp.  57-66,  1897. 

C.  Special  Cases  of  the  Algorithm. 

84.  Hermite.     Surla  fonction  exponentielle.     Comp.  Rend.,  vol.  77,  pp.    -^ 

18-24,  74-79,  226-233,  285-293,  1873. 

This  is  the  famous  work  proving  the  transcendence  of  e. 

85.  Hermite.     (a)  Sur   1'expression    U  sin  x  4-  V  cos  x  +  W.      Extrait 

d'une  lettre  a  Monsieur  Paul  Gordan.     Jour,  fur  Math.,  vol. 
76,  pp.  303-311,  1873. 

(b)  Sur  quelques  approximations  algebriques.     Ibid. ,  vol.  76,  pp.     ^ 
342-344,  1873. 

(c)  Sur  quelques  equations  differentielles  lineaires.     Extrait  d'une 
lettre  a  M.  L.  Fuchs  de  Gottingue.     Ibid. ,  vol.  79,  pp.  324-338, 
1875. 

86.  Laguerre.     Sur  la  fonction  exponentielle.     Bull.delaSoc.Math.de 

France,  vol.  8  (1880),  pp.  11-18,  or  Oeuvres,  vol.  1,  p.  336. 

87.  Humbert,     (a)  Sur  une  generalisation  de  la  theorie  des  fractions 

continues  algebriques.     Bull,  de  la  Soc.  Math,  de  France,  vol. 
8,  pp.  191-196  ;  vol.  9,  pp.  24-30,  1879-1881. 
(6)  Sur  la  fonction  (x—  1)".    Ibid.,  vol.  9,  pp.  56-58,  1880-81. 

88.  Pincherle.     Sulla  rappresentazione  approssimata   di  una  funzione 

mediante  irrazionali  quadratic!.     Rendiconti,  R.  Istituto  Lorn-  -"s^ 
bardo  di  Scienze  e  Lettere,  ser.  2,  vol.  23,  pp.  373-376,  1890. 

89.  Pincherle.     (a)   Una    nuova    estensione    delle    funzioni    sferiche.      ^ 

Memoirie  della  R.  Accad.  delle  Scienze  deU'I>titutodi  Bologna,  """"--s 
ser.  5,  vol.  1,  pp.  337-370,  1890. 

(a7)  Sulla  generalizzazione  delle  funzioni  sferiche.  Bologna  Ren 
diconti,  1891-92,  pp.  31-34. 

(6)  Un  sistema  d'integrali  ellittici  considerati  come  funzioni 
delFinvariante  assoluto.  Atti  della  R.  Accad.  dei  Lincei,  ser. 
4.  vol.  717  pp.  74-80,  1891. 

90.  Bortolotti.    (a)  Sui  sistemi  ricorrenti  del  3°  ordiue  ed  in  particolare 

sui  sistemi  periodici.     Rendiconti  del  Circolo  di  Palermo,  vol.  5, 
pp.  129-151,  1891. 

(6)  Sulla  generalizzazione  delle  frazioni  continue  algebriche  peri-  - 
odiche.     Ibid.,  vol.  6,  pp.  1-13,  1892. 

VI.    Series  of  Polynomial*  (N(iherungsnenner).      9 
The  series 


V  —  U 


182  THE   BOSTON   COLLOQUIUM. 

was  first  given  by  Heine  in  Crelle's  Jour.,  vol.  42  (1851),  p.  72. 
See  also  his  Handbuch,  vol.  1,  pp.  78-79,  197-200.  Among 
the  numerous  works  relating  to  expansions  in  terms  of  Kugel 
functionen  erster  und  zweiter  Gattung  may  be  mentioned  : 

91.  Bauer.     Von  den  Coefficienten  der  Reihen  von  Kugelfunctionen 

einer  Variablen.     Jour,  fur  Math.,  vol.  56,  pp.  101-121,  1859. 

92.  C.  G.  Neumann.     Ueber  die  Entwickelung  einer  Function  mit  imag- 

inarem  Argumente  nach  den  Kugelfunctionen  erster  und  zweiter 
Gattung,  Halle,  1862. 

93.  Thome.     Ueber   die  Reihen   welche    nach   Kugelfunctionen    fort- 

schreiten.     Jour,  fur  Math.,  vol.  66,  pp.  337-343,  1866. 

94.  Laurent.     Memoire  sur  les  fonctions  de  Legendre.     Jour,  de  Math., 

ser.  3,  vol.  1,  pp.  373-398,  1875. 

See  the  comments  by  Heine  in  vol.  2,  pp.  155-157,  also  by 
Darboux  and  Laurent  in  the  same  vol.,  pp.  240,  420. 


Numerous  memoirs  relate  to  series  in  terms  of  the  polynom 
ials  arising  from  the  expansion  of  (1  —  2ax  +  a2)".  It  suffices 
here  to  refer  to  the  Encyklopadie  der  Math.  Wissenschaften, 
I  A  10,  §31. 

95.  Frobenius.     Ueber  die   Entwicklung   analytischer    Functionen   in 
^^^    .  Reihen,  die  nach   gegebenen  Functionen  fortschreiten.     Jour. 

fur  Math.,  vol.  73,  pp.  1-30,  1871. 
An  interesting  memoir. 

96.  Darboux.     Sur  1' approximation  des  fonctions  de  tres-grands  nom- 

bres  et  sur  une  classe  etendue  de  developpements  en  serie,  Part 
2.     Jour,  de  Math.,  ser.  3,  vol.  4,  pp.  377-416,  1878. 

97.  Gegenbauer     Ueber  Kettenbriiche.    Wiener  Berichte,  vol.  80,  Abth. 

2,  pp.  763-775,  1880. 

98.  Poincare.     (a)  Sur  les  equations  lineaires  aux  differentielles  ordi- 

naires  et  aux  differences  finies.     Amer.  Jour,  of  Math.,  vol.  7, 
pp.  243-257,  1885. 

This  gives  an  important  criterion  for  the  convergence  of  series 
of  polynomials.  See  lecture  4. 

(a')  Sur  les  series  des  polynomes.     Comp.  Rend.,  vol.  56,  p.  637, 
1883. 

99.  On  the  series  ^An(x  —  a^(x  —  «2)  •  •  •  (x  —  an}. 

A  series  of  this  form  is  employed  in  Newton's  interpolation 
formula,  Philosophise  naturalis  principia,  book  3,  lemma  V. 
See  the  Encyklopadie  der  Math.  Wissenschafcen,  I  D  3,  §  3.  A 
similar  uj-e  is  made  by 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    183 

Cauchy.     (a)  Sur  les  fonctions  interpolates.     Comp.  Rend.,  vol. 

11,  pp.  775-789,  1841. 
See  next  No.  95. 

Peano.     (6)  Sulle  funzioni  interpolari.     Atti  della  R.  Accad.  delle 

Scienze  di  Torino,  vol.  18,  pp.  573-580,  1883. 

Bendixson.     (c)  Sur  une  extension  a  1'infini  de  la  formule  d' inter 
polation  de  Gauss.     Acta  Math.,  vol.  9,  pp.  1-34,  1886. 
(c')  Sur  la  formule  d' interpolation  de  Lagrange.     Comp.  Rend., 

vol.  101  (1885),  pp.  1050-1053  and  1129-1131 . 
Pincherle.    (d)  Sull'interpolazione.     Memoirie  della  R.  Accad.  delle 
Scienze  di  Bologna,  ser.  5,  vol.  3,  pp.  293-318. 

(See  a   u  note  historique"   by  Enestrom,  Comp.    Rend.,  vol. 
103,  p.  523,  1886). 
See  also  No.  103. 

100.  Pincherle,     Sur   le  developpement  d'une  fonction   analytique    en 

serie  de  polynomes.     Comp.  Rend.,  vol.  107,  p.  986,  1888. 

101.  Pincherle.     Resume  de  quelques  resultats  relatifs  a  la  theorie  des 

systemes  recurrents  de  fonctions.     Mathematical  Papers,  Chi- 
cago  Congress,  1893,  pp.  278-287. 

102.  Blumenthal.     Ueber  die  Entwickelung  einer  willkiirlichen  Funk- 

tion  nach  den  Nennern  des  Kettenbruches  fur 


Dissertation,  Gottingen,  1898. 

The  most  advanced  development  of  this  subject  is  found  in 
the  work  of  Blumenthal  and  Pincherle. 

103.  Laurent.     Sur  les  series  de  polynomes.     Jour,   de  Math.,  ser.    5, 

vol.  8,  pp.  309-328,  1902. 

104.  Stekloff.     Sur    le  developpenient  d'une  fonction  donee  en    series 

procedant  suivant  les  polynomes  de  Tchebicheff  et,  en  particul- 
ier,  suivaut  les  polynomes  de  Jacobi.  Jour,  fur  Math.,  vol. 
125,  pp.  207-236,  1903. 

See  also  Nos.  20,  70.  71. 

104  bis.  Rouche.  Memoire  sur  le  developpemeut  des  fonctions  en  series 
ordonnees  suivant  les  denorninateurs  des  reduites  d'une  frac 
tion  continue.  Jour,  de  FEc  P«»lytech.,  cahier  37,  pp.  1-34. 

This  mem  )ir  has  a  close  connection  with  the  work  of  Tcheby- 
chef. 

VII.   On  the  Roots  of  the  Numerators  and  Denominators  of  the  Convergents. 

105.  Sylvester,     (a)  On  a  remarkable  modification  of  Sturm's  theorem. 

Phil.  Mag.,  ser.  4,  vol.  5,  pp.  446-456,  1853. 


184  THE   BOSTON   COLLOQUIUM. 

(5)  Note  on  a  remarkable  modification  of  Sturm's  theorem  and  on 
a  new  rule  for  finding  superior  and  inferior  limits  to  the  roots  of 
an  equation.  Ibid.,  vol.  6,  pp.  14-20,  1853. 

(c)  On  a  new  rule  for  finding  superior  and  inferior  limits  to  the 
real  roots  of  any  algebraic  equation.     Ibid.,  vol.  6,  pp.  138-140, 
1853. 

(d)  Note  on  the  new  rule  of  limits.     Ibid.,  vol.  6,  pp.   210-213, 
1853. 

(<?)  On  a  theory  of  the  syzygetic  relations  of  two  rational  integral 
functions,  comprising  an  application  to  the  theory  of  Sturm's 
functions,  and  that  of  the  greatest  algebraic  common  measure. 
Phil.  Trans.,  1853  ;  see  in  particular  p.  496  ff. 

(/)  Theoreme  sur  les  limitesdes  racines  reelles  des  equations  alge- 
briques.  Nouvelles  Ann.  de  Math.,  ser.  1,  vol.  12,  pp.  286-287, 
1853. 

(#)  Pour  trouver  une  limite  superieure  et  une  limite  inferieure  des 
racines  reelles  d'une  equation  quelconque.  Ibid.,  ser.  1,  vol.  12, 
pp.  329-336,  1853. 

106.  Laguerre.  Sur  quelques  proprietes  des  equations  algebriques  qui 
ont  toutes  les  racines  reelles.  Nouvelles  Ann.  de  Math.,  ser.  2, 
vol.  19  (1880),  pp.  224-239,  or  Oeuvres,  vol.  1,  pp.  113-118. 

Laguerre  considers  here  the  roots  of  the  numerators  and  de 
nominators  of  the  approximants  for/(x)  and  ljf(x)  when  f(x)  is  a 
polynomial  with  real  roots. 

107.  Gegenbauer.     (a)  Ueber  algebraische  Gleichungen  welche  nur  reele 

Wurzeln  besitzen.     Wiener  Berichte,  vol.    84  (1882),   Abt.  2, 
see  in  particular  pp.  1106-1107. 

(ft)  Ueber  algebraische  Gleichungen  welche  eine  bestimmte  An- 
zahl  complexer  Wulzeln  besitzen.  Ibid,«  vol.  87,  pp.  264-270, 
1883. 

108.  Markoff.     Sur  les  racines   de   certaines  equations.     Math.  Ann., 

vol.  27,  pp.  143-150,  1886. 

108  bis.  Hurwitz.     Ueber   die   Nullstellen   der   Bessel'schen    Function. 
Math.  Ann.,  vol.  33,  pp.  246-266,  1889. 

Although  the  functions  considered  in  this  memoir  are  of  a 
special  character,  the  memoir  is  mentioned  here  on  account  of 
the  methods  employed.  ^» 

109.  Porter.     On  the  roots  of  functions  connected  by  a  linear  recurreW 

relation  of  the  second  order.     Annals  of  Math.,  ser.  2,  vol.  3, 
pp.  55-70,  1902. 
See  also  Nos.  20,  26a,  31,  32a,  45,  56,  71,  74,  76,  87<i,  118a. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    185 

VIII .     Approximation  to  a  Function  at  More  Than  One  Point.     Connection 
of  Continued  Fractions  with  the  Theory  of  Interpolation. 
Under  No.  99   have   been  already  classified   various  works 
which  relate  to  simultaneous  approximation  at  several  points. 
In  addition,  the  following  memoirs  may  also  be  consulted: 

110.  Cauchy.     Sur   la   formule   de   Lagrange    relativ   a   interpolation. 

Analyse  Alg.,  p.  528,  or  Oeuvres,  ser.  2,  vol.  3,  pp.  429-433. 

111.  Jacobi.     Ueber  die  Darstellung  eine  Reihe  gegebner  Werthe  durch 

eine  gebrochene  rationale  Function.     Jour,  fur  Math.,  vol.  30, 
pp.  127-156,  1846.  or  Werke,  vol.  3,  p.  479. 

112.  Fade.     Sur  F  extension  des  proprietes  des  reduites  d'une  fonction 

aux  fractions  d' interpolation  de  Cauchy.     Comp.  Rend.,  vol. 
130,  p.  697,  1900. 
See  also  Nos.  95,  99. 


For  general  works  upon  interpolation  which  bring  out  the 
relation  of  the  subject  to  continued  fractions,  see  Heine's 
Handbuch  der  Kugelfunctionen,  vol.  2,  and  MarkofFs  Differ- 
enzenrechnung  (deutsche  Uebersetzung),  chap.  1,  6,  7  ;  also  the 
following  memoir  : 

113.  Posse.     Sur  quelques   applications  des  fractions   continues   alge- 

briques.     Pp.  1-175,  1886. 

114.  Gauss.     Methodus  nova  integralium  valores  per  approximationem 

inveniendi.     Werke,  vol.  3,  pp.  165-196,  1816. 

115.  Christoffel.      Ueber  die  Gaussische  Quadratur  und  eine  Verallge- 

meinerung  derselben.     Jour,  fur  Math.,  vol.  55,  pp.  61-82, 1858. 

116.  Mehler.     Bemerkungen  zur  Theorie  der  mechanischen  Quadraturen. 

Ibid.,  vol.  63,  pp.  152-157,  1864. 

117.  Posse.     Sur  les  quadratures.     Nouvelles  Ann.  de  Math.,  ser.    2, 

vol.  14,  pp.  49-62,  1875. 

118.  Stieltjes.     (a)  Quelques  recherches  sur  la  theorie  des  quadratures 

dites  mecaniques.     Ann.  de  1'Ec.  Nor.,  ser.  3,  vol.  1,  pp.  409- 
426,  1884. 

We  find  here  the  origin  of  his  notable  1894  memoir,  No.  26a. 
(a7)  Sur   1'evaluation  approchee   des   integrates.     Comp.  Rend., 
vol.  97,  pp.  740  und  798,  1883. 

(b)  Note  sur  1'  integrate  Cf(x)Q(x)dx. 

Jo. 

Nouv.  Ann.  de  Math.,  ser.  3,  vol.  7,  pp.  161-171,  1888. 

119.  Markoff.     Sur  la  methode  de  Gauss  pour  le  calcul  approche  des  in 

tegrales.     Math.  Ann.,  vol.  25,  pp.  427-432,  1885. 


186  THE   BOSTON   COLLOQUIUM. 

120.  Pincherle.     Su  alcune  forme  approssimate  per  la  rappresentazione 

di  funzioni.    MemoiriedellaR.  Accad.  delle  Scienze  dell'Istituto 
di  Bologna,  ser.  4,  vol.  10,  pp.  77-88,  1889. 

121.  Tchebychef.     A  brief  sketch  of  the  memoirs  below  will  be  found  on 

pp.  17-20  of  Vassilief's  memoir  on  "P.  L.  Tchebychef  et  son 
oeuvre  scientifique. " 

(a)  Surles  fractions  continues.    Jour,  de  Math.,  ser.  2,  vol.  3,  pp. 
X       289-323,  1858,  or  Oeuvres,  vol.  1,  p.  203-230. 

(6)  Sur  une  formule  d'analyse.  Bull.  Phys.  Math,  de  1'Acad.  des 
sciences  de  St.  Petersbourg,  vol.  13,  pp.  210-211, 1854,  or  Oeuv 
res,  vol.  1,  pp.  701-702. 

(c)  Sur  une  nouvelle  serie.     Ibid.,  vol.  17,  pp.  257-261,  1858,  or 
Oeuvres,  vol.  1,  pp.  381-384. 

(d)  Sur  1' interpolation  par  la  methode  des  moindres  carres.  Mem. 
de  1'Acad.  des  sciences  de  St.  Petersbourg,  ser.  7,  vol.  1,  pp. 
1-24,  1859,  or  Oeuvres,  vol.  1,  pp.  473-498. 

(e)  Sur  le  developpement  des  fonctions  a  une  seule  variable.    Bull, 
de  PAcad.  imp.  des  sciences  de  St.  Petersbourg,  ser.  7,  vol.  ly 
pp.  J  94-199,  1860,  or  Oeuvres,  vol.  1,  pp.  501-508. 

IX.  MISCELLANEOUS. 

122.  Tchebychef.     (a)  Sur  les  fractions  continues  algebriques.     Jour,  de 

Math.,  ser.  2,  vol.  10,  pp.  353-358,  1865,  or  Oeuvres,  vol.  1,  pp. 
611-614. 

(b)  Sur  le  developpement  des  fonctions  en  series  a  1' aide  des  frac 
tions  continues,  1866.     Oeuvres,  vol.  1,  pp.  617-636. 

(c)  Sur  les  expressions  approchees,  lineares  par  rapport  a  deux 
polynomes.     Bull,  des  sciences  Math,  et  Astron.,  ser.  2,  vol.  1, 
pp.  289,  382  ;  1877. 

Hermite.  (d)  Sur  une  extension  donnee  a  la  theorie  des  fractions 
continues  par  M.  Tchebychef.  Jour,  fur  Math.,  vol.  88,  pp. 
12-13,  1880. 

123.  Tchebychef.     (a)     Sur  les  valeurs  limites  des  integrales.     Jour,  de 

Math.,  ser.  2,  vol.  19,  pp.  157-160,  1874. 

(6)  Sur  la  representation  des  valeurs  limites  des  integrales  par  des 
residus  integraux  (1885).  Acta.  Math.  vol.  9,  pp.  35-56,  1887. 

Markoff.  (c)  Demonstration  de  certaines  inegalites  de  M.  Tcheby 
chef.  Math.  Ann.,  vol.  24,  pp.  172-178,  1884. 

(d)  Nouvelles  applications  des  fractions  continues.     Math.  Ann., 
vol.  47,  pp.  579-597,  1896. 

124.  Laguerre.     (a)  Sur  le  developpement  de  (x  —  z)m  suivant  les  puis 

sances   de   (z2  —  1).     Comp.    Kend.,    vol.  86  (1878),  p.  956,  or 
Oeuvres,  vol.  1,  p.  315. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.    187 

(6)  Sur  le  developpement  d'une  fonction  suivant  les  puissances 
d'une  polynome.  Jour,  fur  Math.,  vol.  88  (1880) ;  in  particular, 
p.  37,  or  Oeuvres,  vol.  1,  p.  298. 

(c)  Same  subject.     Comp.  Rend.,  vol.  86,  (1878)  p.  383,  or  Oeuv 
res,  vol.  1,  p.  295. 

(d)  Sur  quelques  theoremes  de  M.  Hermite.     Extrait  d'une  lettre 
addressee  a  M.  Borchardt.     Jour,  fur  Math.,  vol.  89  (1880),  pp. 
340-342,  or  Oeuvres,  vol.  1,  p.  360. 

125.  Sylvester.     Preuve  que  TT  ne  peut  pas  etre  racine  d'une  equation 

algebrique  a  coefficients  entiers.     Comp.  Rend.,  vol.  Ill,  pp. 
866-871,  1890. 

A  fundamental  error  in  the  proof  has  been  pointed  out  by 
Markoff.     See  p.  386  of  vol.  30  of  the  Fortschritte  der  Math. 

126.  Gegenbauer.     Ueber  die  Naherungsnenner  regularer  Kettenbriiche. 

Monatshefte  far  Math,  und  Phys.,  vol.  6,  pp.  209-219,  1895. 

127.  Bortolotti.     Sulla  rappresentazione  approssimata  di  funzioni  alge- 

briche  per  mezzo  di  funzioni  razionale.     Atti  della  R.  Accad. 
dei  Lincei,  ser.  5,  vol.  11?  pp.  57-64,  1899. 

ADDENDUM  TO  I  A. 

128.  Euler.    De  fractionibus  continuis  dissertatio.   Comment.    Petrop., 

vol.  9,  p.  129  ff.,  1737. 


RETURN    Astronomy/Mathematics/Statistics/Computer  Science  Library 

1  00  Evans  Hall  642-3381 


LOAN  PERIOD  1 
7  DAYS 

2                               3 

4 

5                               6 

ALL  BOOKS  MAY  BE  RECALLED  AFTER  7  DAYS 

DUEAS  STAMPED  BELOW 

06  f  ^  t>\€&  ^0 

ICS^ap^j 

X 

wySs 

UNIVERSITY  OF  CALIFORNIA,  BERKELEY 

FORM  NO.  DD3,  5m,  3/80  BERKELEY,  CA  94720 

®$ 


IWOH. 


